text: The best sensitivity we can get for the LVC comes from considering ultra-high-energy cosmic-ray protons, which are seen to arrive on Earth with energies above several tens of EeV’s from more or less all directions [36]. This means that we can obtain a (one-sided) sensitivity for kµν of (cid:46) − r kµνpˆ pˆ 10 23 . (28) 1 µ µ This has the potential of improving at least some of the limits in Tab. I. Notice that actually obtaining limits from Cherenkov-like processes requires some the- oretical work. Strictly speaking, one has to calculate the actual emission rate and demon- strate that it does not vanish. But more importantly, one has to verify that the theory with spacelike momenta can be made consistent, since spacelike momenta correspond to negative energies in some observer frames (notably the restframe of the decaying particle). For a diﬀerent LVC it has been shown that such a theory can nevertheless be quantized and does not contain runaway stability issues [12]. For the present parameter this remains to be shown, however. B. W µνρσ and the nucleon-nucleon potential The fully traceless part of kµνρσ does not appear in any kinetic term for baryons or G light mesons. The lowest-order term allowed by all the symmetries is the one in Eq. (15d), which is a pion-nucleon interaction term. A similar situation was identiﬁed in Ref. [8] for a diﬀerent LVC. As in Ref. [8], we can calculate the nucleon-nucleon potential that follows from considering one-pion exchange between the two nucleons, with one of the vertices originating form the LV πN operator in Eq. (15d). The resulting potential is given by −8g · (σiσm + σmσi)qjqm V = A (r + r )W 0i0jτ τ 1 2 1 2 , (29) LV F 3 4 1 2 q2 + m2 0 π where σ (τ ) are spin (isospin) operators corresponding the interacting nucleons 1,2 1,2 − (cid:48) and q = p p is the momentum transfer that ﬂows from nucleon 1 to nucleon 2, while p and (cid:48) p are the relative momenta of the incoming and outgoing nucleon pair in the center-of-mass frame. As usual, the Latin indices run only over spatial directions. 17 ##LTLine## ##LTLine##

text: The

text: best

text: sensitivity

text: we

text: can

text: get

text: for

text: the

text: LVC

text: comes

text: from

text: considering

text: ultra-high-energy

text: cosmic-ray

text: protons,

text: which

text: are

text: seen

text: to

text: arrive

text: on

text: Earth

text: with

text: energies

text: above

text: several

text: tens

text: of

text: EeV’s

text: from

text: more

text: or

text: less

text: all

text: directions

text: This

text: means

text: that

text: we

text: can

text: obtain

text: a

text: (one-sided)

text: sensitivity

text: for

text: kµν

text: of

text: r

text: kµνpˆ

text: pˆ

text: µ

text: µ

text: This

text: has

text: the

text: potential

text: of

text: improving

text: at

text: least

text: some

text: of

text: the

text: limits

text: in

text: Tab.

text: I.

text: Notice

text: that

text: actually

text: obtaining

text: limits

text: from

text: Cherenkov-like

text: processes

text: requires

text: some

text: the-

text: oretical

text: work.

text: Strictly

text: speaking,

text: one

text: has

text: to

text: calculate

text: the

text: actual

text: emission

text: rate

text: and

text: demon-

text: strate

text: that

text: it

text: does

text: not

text: vanish.

text: But

text: more

text: importantly,

text: one

text: has

text: to

text: verify

text: that

text: the

text: theory

text: with

text: spacelike

text: momenta

text: can

text: be

text: made

text: consistent,

text: since

text: spacelike

text: momenta

text: correspond

text: to

text: negative

text: energies

text: in

text: some

text: observer

text: frames

text: (notably

text: the

text: restframe

text: of

text: the

text: decaying

text: particle).

text: For

text: a

text: diﬀerent

text: LVC

text: it

text: has

text: been

text: shown

text: that

text: such

text: a

text: theory

text: can

text: nevertheless

text: be

text: quantized

text: and

text: does

text: not

text: contain

text: runaway

text: stability

text: issues

text: For

text: the

text: present

text: parameter

text: this

text: remains

text: to

text: be

text: shown,

text: however.

text: B.

text: W

text: µνρσ

text: and

text: the

text: nucleon-nucleon

text: potential

text: The

text: fully

text: traceless

text: part

text: of

text: kµνρσ

text: does

text: not

text: appear

text: in

text: any

text: kinetic

text: term

text: for

text: baryons

text: or

text: G

text: light

text: mesons.

text: The

text: lowest-order

text: term

text: allowed

text: by

text: all

text: the

text: symmetries

text: is

text: the

text: one

text: in

text: Eq.

text: which

text: is

text: a

text: pion-nucleon

text: interaction

text: term.

text: A

text: similar

text: situation

text: was

text: identiﬁed

text: in

text: Ref.

text: for

text: a

text: diﬀerent

text: LVC.

text: As

text: in

text: Ref.

text: we

text: can

text: calculate

text: the

text: nucleon-nucleon

text: potential

text: that

text: follows

text: from

text: considering

text: one-pion

text: exchange

text: between

text: the

text: two

text: nucleons,

text: with

text: one

text: of

text: the

text: vertices

text: originating

text: form

text: the

text: LV

text: πN

text: operator

text: in

text: Eq.

text: The

text: resulting

text: potential

text: is

text: given

text: by

text: (σiσm

text: σmσi)qjqm

text: V

text: A

text: (r

text: r

text: )W

text: τ

text: LV

text: F

text: π

text: where

text: σ

text: (τ

text: are

text: spin

text: (isospin)

text: operators

text: corresponding

text: the

text: interacting

text: nucleons

text: and

text: q

text: p

text: p

text: is

text: the

text: momentum

text: transfer

text: that

text: ﬂows

text: from

text: nucleon

text: to

text: nucleon

text: while

text: p

text: and

text: p

text: are

text: the

text: relative

text: momenta

text: of

text: the

text: incoming

text: and

text: outgoing

text: nucleon

text: pair

text: in

text: the

text: center-of-mass

text: frame.

text: As

text: usual,

text: the

text: Latin

text: indices

text: run

text: only

text: over

text: spatial

text: directions.

text: ##LTLine##

text: ##LTLine##


text: [27] E. E. Jenkins and A. V. Manohar, Phys. Lett. B 255, 558 (1991); V. Bernard, N. Kaiser, J. Kambor and U. G. Meissner, Nucl. Phys. B 388, 315 (1992). [28] S. R. Coleman and S. L. Glashow, Phys. Lett. B 405, 249 (1997). [29] A. Kostelecky´ and M. Mewes, Phys. Rev. D 88, 096006 (2013). [30] P. Wolf, F. Chapelet, S. Bize and A. Clairon, Phys. Rev. Lett. 96, 060801 (2006). [31] M. Smiciklas, J. M. Brown, L. W. Cheuk and M. V. Romalis, Phys. Rev. Lett. 107, 171604 (2011). [32] One could argue that there is also a contribution coming from a W -boson loop. This would (cid:39) − involve a contribution from kµνρσ and would be suppressed by a factor q4/M4 10 12, W W (cid:39) − where q 10 4 GeV is the typical energy of photons coming from GRBs. Additionally, there are contributions from a mass-dimension three CPT-odd coeﬃcient. However, for this LVC the birefringence is energy independent and therefore the studies of GRBs are not sensitive to it. This CPT-odd coeﬃcient has been bounded by looking at the isotropy of the CMB − polarization and determined to be of order kµ < 10 43 GeV [6]. AF [33] V. A. Kostelecky´ and M. Mewes, Phys. Rev. D 80, 015020 (2009); V. A. Kostelecky´ and M. Mewes, Phys. Rev. Lett. 110, 201601 (2013). [34] M. de Cesare, N. E. Mavromatos and S. Sarkar, Eur. Phys. J. C 75, no. 10, 514 (2015). [35] J. P. Noordmans and K. K. Vos, Phys. Rev. D 89, 101702 (2014); B. Altschul, Phys. Rev. D 77, 105018 (2008); Phys. Rev. D 93, 105007 (2016). [36] A. Aab et al. [Pierre Auger Collaboration], Astrophys. J. 804, no. 1, 15 (2015); T. Abu-Zayyad et al. [Telescope Array Collaboration], Astrophys. J. 777, 88 (2013). [37] S. K. Peck, D. K. Kim, D. Stein, D. Orbaker, A. Foss, M. T. Hummon and L. R. Hunter, Phys. Rev. A 86, 012109 (2012). [38] D. Eversmann et al. [JEDI Collaboration], Phys. Rev. Lett. 115, 094801 (2015). 21

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text: Manohar,

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text: Wolf,

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text: Chapelet,

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text: M.

text: Smiciklas,

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text: W.

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text: and

text: M.

text: V.

text: Romalis,

text: Phys.

text: Rev.

text: Lett.

text: One

text: could

text: argue

text: that

text: there

text: is

text: also

text: a

text: contribution

text: coming

text: from

text: a

text: W

text: -boson

text: loop.

text: This

text: would

text: involve

text: a

text: contribution

text: from

text: kµνρσ

text: and

text: would

text: be

text: suppressed

text: by

text: a

text: factor

text: W

text: W

text: where

text: q

text: GeV

text: is

text: the

text: typical

text: energy

text: of

text: photons

text: coming

text: from

text: GRBs.

text: Additionally,

text: there

text: are

text: contributions

text: from

text: a

text: mass-dimension

text: three

text: CPT-odd

text: coeﬃcient.

text: However,

text: for

text: this

text: LVC

text: the

text: birefringence

text: is

text: energy

text: independent

text: and

text: therefore

text: the

text: studies

text: of

text: GRBs

text: are

text: not

text: sensitive

text: to

text: it.

text: This

text: CPT-odd

text: coeﬃcient

text: has

text: been

text: bounded

text: by

text: looking

text: at

text: the

text: isotropy

text: of

text: the

text: CMB

text: polarization

text: and

text: determined

text: to

text: be

text: of

text: order

text: kµ

text: GeV

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text: M.

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text: Phys.

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text: Phys.

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text: et

text: al.

text: [Pierre

text: Auger

text: Collaboration],

text: Astrophys.

text: J.

text: no.

text: T.

text: Abu-Zayyad

text: et

text: al.

text: [Telescope

text: Array

text: Collaboration],

text: Astrophys.

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text: S.

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text: Peck,

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text: Collaboration],

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text: Lett.


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text: formulas for the two cases can be obtained using respectively the T M or T E reﬂection = + = + = + = + coeﬃcients. D−eﬁning (no summati−on over repeated indices)− − ( ) Eijl ri,jrj,le 2djKj 1, F ijl ri,je 2djKj rj,l, Gijl ri,j e 2djKjrj,l, Hijl e 2djKj ri,jrj,l, = we have for the single cavity 012 in Fig. 1 ( ) ∆ 1 E012; (2) ( − ) = + =∶ + for two cavities (0)12 234 : − ( ) ( ) ⎛ ⎞ = ( ) + + ( ) ∆(2) E012E234 e 2 d2k2 F⎝012G234 E01⎠2E234 I 2 and (3) 012 I 2 log ∆ 2 log E012E234 log 1 012 , (4) E012E234 and for the three cavities: = + ( ) − ( + ) + + − − ∆ 3 E=∶012E234E456 e+2 d2k2+d4k4 F 012H+ 234G456 ( ) ( ) ( ) e 2d2k2E456F 012G234 e 2d4k4E012F 234G456 = ( ) + ( + ) ( ) ( ) E012E234E456 I 3 E456I 2 E012I 2 , (5) 012 234 ⎛ + ⎞ I 3 log ∆ 3 l+og E012+E234E45(6) log 1( ) ⎝ + E⎠012E234E456 ( ) E456I 2 E012I 2 log 1 012 234 . (6) E012E234E456 I 3 → ∞ → ( ) = = + In this way, when d ([se)e Eq. (3)] I 2 0 and 2 012 log ∆ 2 log E012E234 log E012 log E234. ( ) − ( ) ( ) That is to say, when the two cavities are far away their energy is simply the sum of the individual co=ntrib=utions.=In this=resp=ect the=seco=nd term on the righ=t of Eq. (3), =I 2 , can be seen as the energy due to the coupling of the two cavities 012 234 . When d d d , d d , (cid:15) (cid:15) (cid:15) , (cid:15) (cid:15) (cid:15) we obtain E012 E234, F 012 F 234 so 1 3 5 2 4 0 2 4 1 3 5 TM that we can omit the subscripts: = ( ) + ( + ) ( ) ( ) ( ) ( ) ( ) I 2 2 log ∆ 2 log ∆ 1 log 1 . (7) ( − − ) 2 ∆ 1 For the three cavities 012 234 456 , formulas are written so as to make evident ( − ) ( − ) the contribution to the energy resulting from the sum of the energies of the single cavity, ( ) with respect to the one coming from the coupling of the two possible pairs of cavities 012 234 , 234 456 , and the one coming from the coupling of the three: I 3 . Thus, 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: formulas

label: paragraph

text: for

text: the

text: two

text: cases

text: can

text: be

text: obtained

text: using

text: respectively

text: the

text: T

text: M

text: or

text: T

text: E

text: reﬂection

text: coeﬃcients.

text: D−eﬁning

text: (no

text: summati−on

text: over

text: repeated

text: indices)−

text: Eijl

text: ri,jrj,le

text: F

text: ijl

text: ri,je

text: rj,l,

text: Gijl

text: ri,j

text: e

text: Hijl

text: e

text: ri,jrj,l,

label: equation

text: we

text: have

text: for

text: the

text: single

text: cavity

text: in

text: Fig.

text: for

text: two

text: cavities

text: e

text: I

text: and

text: I

text: log

text: log

text: log

text: and

text: for

text: the

text: three

text: cavities:

text: F

text: e

text: e

text: I

text: I

text: log

text: l+og

text: log

text: log

text: I

text: In

text: this

text: way,

text: when

text: d

text: ([se)e

text: Eq.

text: I

text: and

text: log

text: log

text: log

text: log

text: That

text: is

text: to

text: say,

text: when

text: the

text: two

text: cavities

text: are

text: far

text: away

text: their

text: energy

text: is

text: simply

text: the

text: sum

text: of

text: the

text: individual

text: co=ntrib=utions.=In

text: this=resp=ect

text: the=seco=nd

text: term

text: on

text: the

text: righ=t

text: of

text: Eq.

text: =I

text: can

text: be

text: seen

text: as

text: the

text: energy

text: due

text: to

text: the

text: coupling

text: of

text: the

text: two

text: cavities

text: When

text: d

text: d

text: d

text: d

text: d

text: we

text: obtain

text: F

text: F

text: so

text: TM

text: that

text: we

text: can

text: omit

text: the

text: subscripts:

text: I

text: log

text: log

text: log

text: For

text: the

text: three

text: cavities

text: formulas

text: are

text: written

text: so

text: as

text: to

text: make

text: evident

text: the

text: contribution

text: to

text: the

text: energy

text: resulting

text: from

text: the

text: sum

text: of

text: the

text: energies

text: of

text: the

text: single

text: cavity,

text: with

text: respect

text: to

text: the

text: one

text: coming

text: from

text: the

text: coupling

text: of

text: the

text: two

text: possible

text: pairs

text: of

text: cavities

text: and

text: the

text: one

text: coming

text: from

text: the

text: coupling

text: of

text: the

text: three:

text: I

text: Thus,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 where Z is a constant related to the algebra. This equation shows that the gauge variation of the action depends on the rank-N ﬁeld-strength. The action above is not actually directly zero, and there emerges an anomaly. Now, we use properties of (cid:15) and rewrite equation (18) as below: (cid:90) | δS = CGG D+1 − − − × Z W(D, N)(cid:15)χ1χ2..χD/2(Fχ1 ∂J51...∂χ1 4RJ11...J41) MD+N − − − − (Fχ2 ∂J52...∂χ2 4RJ12...J42)...(FχD/2 ∂J5D/2...∂χD/2 4RJ1D/2...J4D/2) (19) In equation (19), χ, (cid:15)χ1χ2..χD/2 and W(D, N) can be obtained as: χ = Ji...Ji i 1 N (cid:15)I1I2..ID(cid:15)I1I2 (cid:15)I3I4 ...(cid:15)ID−1ID = J11J21...JN1 J12J22...JN2 J1D/2J2D/2...JND/2 (cid:15)I1I2..ID(cid:15)I1I2(cid:15)J11J21...JN1 (cid:15)I3I4(cid:15)J12J22...JN2 ...(cid:15)ID−1ID(cid:15)J1D/2J2D/2...JND/2 = W(D, N)(cid:15)χ1χ(cid:16)2..χD/2 (cid:17)(cid:16) (cid:17) − − (D + 2)(D 2) − D − (D + 2)(D 2) − − D − D − D − − W(D, N) = [ N( 1)][ 1 N( 1)]...1 [N( 1)][N( 1) 1]...1 U(2(0δ)) 8 2 8 2 2 2 | where U is a function of the generalized Kronecker delta. On the other hand, δS has been added to the CGG D+1 main action of supergravity to remove its anomaly. Thus, we can write: (cid:16) (cid:17) | − | − | δS = δS − = Sanomaly− = 0 CGG D+1 Bosonic SU−GRA D+1 Bosonic SUGRA−D+1 ⇒(cid:16) (D + 2)(D 2(cid:17)) − D − (D + 2)(D 2) − − D − × W(D, N) = [ N( 1)][ 1 N( 1)]...1 8 2 8 2 D − D − − [N( 1)][N( 1) 1]...1 U = 0 2 2 − ⇒ ≤ (D + 2)(D 2) N − (21) 8(D 1) 2 This equation indicates that, for a (D + 1)-dimensional space-time, the dimension of the Lie-algebra should be equal or less than a critical value. Under these conditions, the Chern-Sions gravity is free from anomalies and we do not need an extra manifold. On the other hand, as we show in (15), the dimension of the algebra determines the dimension of the ﬁeld-strength. This means that, for a Lie-N-algebra, ﬁeld-strengths should have at most N indices. For example, for a manifold with 11 dimensions, the algebra can be of order three as predicted in recent papers [11, 12] and ﬁeld-strengths may have three indices. IV. A CHERN-SIMONS MANIFOLD BETWEEN TWO 11-DIMENSIONAL MANIFOLDS IN AN 11+3 DIMENSIONAL SPACE-TIME In the previous Section, we have found that, for an eleven-dimensional manifold, the suitable algebra which removes the anomaly in Chern-Simons gravity is a three-dimensional Lie algebra. This means that the rank of the ﬁelds can be of order two or three. However, equation (2) shows that the rank of the ﬁelds may be higher than three in eleven- dimensional supergravity. Thus, in Chern-Simons gravity theory which lives on an eleven-dimensional manifold, some extra anomalies are expected to show up. To remove them, we assume that there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the ﬁrst one by exchanging Chern-Simons ﬁelds. These ﬁelds produce a Chern-Simons manifold that connects these two eleven-dimensional manifolds ( see Fig 1.) Thus, in this model, we have two GG terms which live on 11-dimensional manifolds (see equation (2)) and two CGG terms in the bulk so that each of them interacts with one of the 11-dimensional manifolds. We can write the supergravity in 14-dimensional space-time as follows: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: Z

text: is

text: a

text: constant

text: related

text: to

text: the

text: algebra.

text: This

text: equation

text: shows

text: that

text: the

text: gauge

text: variation

text: of

text: the

text: action

text: depends

text: on

text: the

text: rank-N

text: ﬁeld-strength.

text: The

text: action

text: above

text: is

text: not

text: actually

text: directly

text: zero,

text: and

text: there

text: emerges

text: an

text: anomaly.

text: Now,

text: we

text: use

text: properties

text: of

text: and

text: rewrite

text: equation

text: as

text: below:

text: δS

text: CGG

text: Z

text: W(D,

text: MD+N

text: In

text: equation

text: χ,

text: and

text: W(D,

text: N)

text: can

text: be

text: obtained

text: as:

text: χ

text: Ji...Ji

text: i

text: N

text: W(D,

text: (D

text: D

text: (D

text: D

text: D

text: D

text: W(D,

text: N)

text: N(

text: N(

text: [N(

text: where

text: U

text: is

text: a

text: function

text: of

text: the

text: generalized

text: Kronecker

text: delta.

text: On

text: the

text: other

text: hand,

text: δS

text: has

text: been

text: added

text: to

text: the

text: CGG

text: main

text: action

text: of

text: supergravity

text: to

text: remove

text: its

text: anomaly.

text: Thus,

text: we

text: can

text: write:

text: δS

text: δS

text: Sanomaly−

text: CGG

text: Bosonic

text: SU−GRA

text: Bosonic

text: (D

text: D

text: (D

text: D

text: W(D,

text: N)

text: N(

text: N(

text: D

text: D

text: [N(

text: U

text: (D

text: N

text: This

text: equation

text: indicates

text: that,

text: for

text: a

text: (D

text: space-time,

text: the

text: dimension

text: of

text: the

text: Lie-algebra

text: should

text: be

text: equal

text: or

text: less

text: than

text: a

text: critical

text: value.

text: Under

text: these

text: conditions,

text: the

text: Chern-Sions

text: gravity

text: is

text: free

text: from

text: anomalies

text: and

text: we

text: do

text: not

text: need

text: an

text: extra

text: manifold.

text: On

text: the

text: other

text: hand,

text: as

text: we

text: show

text: in

text: the

text: dimension

text: of

text: the

text: algebra

text: determines

text: the

text: dimension

text: of

text: the

text: ﬁeld-strength.

text: This

text: means

text: that,

text: for

text: a

text: Lie-N-algebra,

text: ﬁeld-strengths

text: should

text: have

text: at

text: most

text: N

text: indices.

text: For

text: example,

text: for

text: a

text: manifold

text: with

text: dimensions,

text: the

text: algebra

text: can

text: be

text: of

text: order

text: three

text: as

text: predicted

text: in

text: recent

text: papers

text: and

text: ﬁeld-strengths

text: may

text: have

text: three

text: indices.

text: IV.

text: A

text: CHERN-SIMONS

text: MANIFOLD

text: BETWEEN

text: TWO

text: MANIFOLDS

text: IN

text: AN

text: DIMENSIONAL

text: SPACE-TIME

text: In

text: the

text: previous

text: Section,

text: we

text: have

text: found

text: that,

text: for

text: an

text: eleven-dimensional

text: manifold,

text: the

text: suitable

text: algebra

text: which

text: removes

text: the

text: anomaly

text: in

text: Chern-Simons

text: gravity

text: is

text: a

text: three-dimensional

text: Lie

text: algebra.

text: This

text: means

text: that

text: the

text: rank

text: of

text: the

text: ﬁelds

text: can

text: be

text: of

text: order

text: two

text: or

text: three.

text: However,

text: equation

text: shows

text: that

text: the

text: rank

text: of

text: the

text: ﬁelds

text: may

text: be

text: higher

text: than

text: three

text: in

text: eleven-

text: dimensional

text: supergravity.

text: Thus,

text: in

text: Chern-Simons

text: gravity

text: theory

text: which

text: lives

text: on

text: an

text: eleven-dimensional

text: manifold,

text: some

text: extra

text: anomalies

text: are

text: expected

text: to

text: show

text: up.

text: To

text: remove

text: them,

text: we

text: assume

text: that

text: there

text: is

text: another

text: manifold

text: in

text: the

text: space-time

text: which

text: interacts

text: with

text: the

text: ﬁrst

text: one

text: by

text: exchanging

text: Chern-Simons

text: ﬁelds.

text: These

text: ﬁelds

text: produce

text: a

text: Chern-Simons

text: manifold

text: that

text: connects

text: these

text: two

text: eleven-dimensional

text: manifolds

text: see

text: Fig

text: Thus,

text: in

text: this

text: model,

text: we

text: have

text: two

text: GG

text: terms

text: which

text: live

text: on

text: manifolds

text: (see

text: equation

text: and

text: two

text: CGG

text: terms

text: in

text: the

text: bulk

text: so

text: that

text: each

text: of

text: them

text: interacts

text: with

text: one

text: of

text: the

text: manifolds.

text: We

text: can

text: write

text: the

text: supergravity

text: in

text: space-time

text: as

text: follows:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 FIG. 1: Two eleven dimensional manifolds + Chern-Simons manifold in 14-dimensional space-time. (cid:90) (cid:16) (cid:90) (cid:90) (cid:90) (cid:90) (cid:17) (cid:16) (cid:90) (cid:90) (cid:90) (cid:90) (cid:17) S − = GG + C¯GG¯ + CG¯G + G¯G¯ = SUGRA 14 MN=3 M11 M11 M11 M11 CGG + C¯GG¯ + CG¯G + C¯G¯G¯ (22) M14 M14 M14 M14 In this equation, CGG and C¯G¯G¯ are related to the Chern-Simons gravities which live on the two eleven-dimensional manifolds and are extracting from GG and G¯G¯ terms. Also, C¯GG¯ and CG¯G correspond to the Chern-Simon ﬁelds which are exchanged between the two manifolds in 14-dimensional space-time. By generalizing the results of (3, 6 and 11), we get: ∼ − − G (F F R R ) + ∂ φ∂ φR ... IJKL IJ KL IJ KL I J KL ∼ − − − 1 δCIiIjIk (cid:15)Iktr(FIiIj RIiIj + ∂Iiφ∂Ijφ 2(cid:15)IiIjIkImφRIiIjIkIm + ...) ∼ − − G¯ (F¯ F¯ R¯ R¯ ) + ∂ φ¯∂ φ¯R¯ ...... IJKL IJ KL IJ KL I J KL ∼ − − − 1 δCIiIjIk (cid:15)Iktr(F¯IiIj R¯IiIj + ∂Iiφ¯∂Ijφ¯ 2(cid:15)IiIjIkImR¯IiIjIkIm + ...) (23) Here, the F’s, R’s and φ’s live on one of the supergravity manifolds as depicted in Figure 1, whereas the F¯’s, R¯’s and φ¯’s are ﬁelds of the other supergravity manifold. To obtain their relations, we should make use of equation (12) and the gauge variation of the action(cid:16)s(cid:90)(22); in so do(cid:90)ing, we obtain(cid:90): (cid:90) (cid:17) (cid:90) (cid:16) (cid:17) δS − = δ CGG + C¯GG¯ + CG¯G + C¯G¯G¯ = SUGRA 14 M14 M14 M14 M14 (cid:90) − (cid:16) − Σ5 trFn trRn + tr(FnR5 n) n=1 M14 (cid:90) − (cid:16) − − − + Σ5 Σn Σn J trF¯JFm trR¯JRm + tr(F¯JR¯5 JFmR5 m) n=1 J=0 m=0 M14 (cid:90) (cid:16) − − (cid:17) − − + Σ5 Σn Σn J trF¯mFJ trR¯mRJ + tr(F¯mR¯5 mFJR5 J) n=1 J=0 m=0 M14 (cid:90) (cid:16) − − (cid:17) + Σ5 trF¯n trR¯n + tr(F¯nR¯5 n) + n=1 M14 − − Σ5n=1(tr(Fn(∂Iiφ∂Ijφ)5 n) + tr(Fn((cid:15)IiIjIkImRIiIjIkImφ)5 n) + M10+MN=3 ##LTFigure## ##LTLine## ##LTLine##

text: FIG.

text: Two

label: caption

text: eleven

text: dimensional

text: manifolds

text: Chern-Simons

text: manifold

text: in

text: space-time.

text: S

text: GG

text: C¯GG¯

text: CG¯G

text: G¯G¯

text: SUGRA

text: CGG

text: C¯GG¯

text: CG¯G

text: C¯G¯G¯

text: In

text: this

text: equation,

text: CGG

text: and

text: C¯G¯G¯

text: are

text: related

text: to

text: the

text: Chern-Simons

text: gravities

text: which

text: live

text: on

text: the

text: two

text: eleven-dimensional

text: manifolds

text: and

text: are

text: extracting

text: from

text: GG

text: and

text: G¯G¯

text: terms.

text: Also,

text: C¯GG¯

text: and

text: CG¯G

text: correspond

text: to

text: the

text: Chern-Simon

text: ﬁelds

text: which

text: are

text: exchanged

text: between

text: the

text: two

text: manifolds

text: in

text: space-time.

text: By

text: generalizing

text: the

text: results

text: of

text: and

text: we

text: get:

text: G

text: (F

text: F

text: R

text: R

text: φ∂

text: φR

text: IJKL

text: IJ

text: KL

text: IJ

text: KL

text: I

text: J

text: KL

text: δCIiIjIk

text: RIiIj

text: ∂Iiφ∂Ijφ

text: G¯

text: (F¯

text: F¯

text: R¯

text: R¯

text: φ¯∂

text: φ¯R¯

text: IJKL

text: IJ

text: KL

text: IJ

text: KL

text: I

text: J

text: KL

text: δCIiIjIk

text: R¯IiIj

text: ∂Iiφ¯∂Ijφ¯

text: Here,

text: the

text: F’s,

text: R’s

text: and

text: φ’s

text: live

text: on

text: one

text: of

text: the

text: supergravity

text: manifolds

text: as

text: depicted

text: in

text: Figure

text: whereas

text: the

text: F¯’s,

text: R¯’s

text: and

text: φ¯’s

text: are

text: ﬁelds

text: of

text: the

text: other

text: supergravity

text: manifold.

text: To

text: obtain

text: their

text: relations,

text: we

text: should

text: make

text: use

text: of

text: equation

text: and

text: the

text: gauge

text: variation

text: of

text: the

text: in

text: so

text: we

text: δS

text: δ

text: CGG

text: C¯GG¯

text: CG¯G

text: C¯G¯G¯

text: SUGRA

text: Σ5

text: trFn

text: trRn

text: n)

text: Σ5

text: Σn

text: Σn

text: J

text: trF¯JFm

text: trR¯JRm

text: m)

text: Σ5

text: Σn

text: Σn

text: J

text: trF¯mFJ

text: trR¯mRJ

text: J)

text: Σ5

text: trF¯n

text: trR¯n

text: n)

text: n)

text: n)

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 10 (cid:90) (cid:90) (cid:90) C¯GG¯ = M11+MN=3 (cid:90) (cid:90) ∂X¯I − − − − − k (δ11(y) + Σ6 Σn ( )6 n(F¯m R¯m + ...)(Fn m Rn m + ...)) = n=1 m=0 ∂y¯I MN=3 M11 ∂X¯I − { } − { } k (δ11(y) + Σ6 Σn ( )6 n( X¯Ii, X¯Ij )n m( XIi, XIj )m) (27) n=1 m=0 ∂y¯I MN=3 M11 (cid:90) (cid:90) (cid:90) CG¯G = M11+MN=3 (cid:90) (cid:90) ∂X¯I − − − − − k (δ11(y) + Σ6 Σn ( )6 n(F¯n m R¯n m + ...)(Fm Rm + ...)) = n=1 m=0 ∂y¯I MN=3 M11 ∂X¯I − { } { } − k (δ11(y) + Σ6 Σn ( )6 n( X¯Ii, X¯Ij )m( XIi, XIj )n m) (28) n=1 m=0 ∂y¯I MN=3 M11 Using the results in equations (24) and substituting equations (25,26,27and 28) in equation (24), we obtain: ∂XI ∂X¯I (cid:90)→ (cid:90) (cid:16) (cid:16) (cid:17) (cid:17) ( ) = ( ) ∂yI ∂y¯I ∂XI − { } { } n S − = k δ11(y) + Σ6 ( )6 n XIi, XIj + X¯Ii, X¯Ij (29) SUGRA 14 n=1 ∂yI MN=3 M11 { } −{ } On the other hand, results in equation (24) show that ( XIi, XIj = X¯Ii, X¯Ij ). Thus, we can conclude that the action given above tends to an action on the three-dimensional manifold. (cid:90) (cid:90) (cid:16) (cid:17) { } −{ } XIi, XIj = X¯Ii, X¯Ij (cid:90) S − = k δ11(y) = SUGRA 14 MN=3 M11 → 4πR3 k k (30) 3 MN=3 where R is the radius of manifold. By redeﬁning scalar strings, we are able to show that the action of supergravity shrinks to one. → 4πR3 − X (k ) 1/11X 3 (cid:90) → 4πR3 − ⇒ X¯ (k ) 1/11X¯ 3 4πR3 − → 4πR3 − 4πR3 S − = (k ) 1k (k ) 1k = 1 (31) SUGRA 14 3 3 3 MN=3 This equation yields results that deserves our comments. In fact, two eleven-dimensional manifolds and the bulk interact with each other via diﬀerent types of C, G and Chern-Simons-ﬁelds. When we sum over supergravities that live on these manifolds and consider ﬁelds in the space between them, we get supergravity in 14-dimensional space. By canceling the anomaly in this new supergravity, we can obtain the relations between ﬁelds. By substituting these relations into the action of the 14-dimensional supergravity, we simply obtain one. This means that the 14-dimensional manifold with all its matter content can be topologically shrunk to one point (See Fig.2.). In fact, the system of the world began from this point and then expand and construct a 14-dimensional world similar to what happens in a Big Bang theory. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C¯GG¯

text: ∂X¯I

text: k

text: Σ6

text: Σn

text: n(F¯m

text: R¯m

text: ...)(Fn

text: m

text: Rn

text: m

text: ∂y¯I

text: ∂X¯I

text: k

text: Σ6

text: Σn

text: n(

text: X¯Ii,

text: X¯Ij

text: )n

text: m(

text: XIi,

text: XIj

text: )m)

text: ∂y¯I

text: CG¯G

text: ∂X¯I

text: k

text: Σ6

text: Σn

text: n(F¯n

text: m

text: R¯n

text: m

text: ...)(Fm

text: Rm

text: ∂y¯I

text: ∂X¯I

text: k

text: Σ6

text: Σn

text: n(

text: X¯Ii,

text: X¯Ij

text: )m(

text: XIi,

text: XIj

text: )n

text: m)

text: ∂y¯I

text: Using

text: the

text: results

text: in

text: equations

text: and

text: substituting

text: equations

text: in

text: equation

text: we

text: obtain:

text: ∂XI

text: ∂X¯I

text: ∂yI

text: ∂y¯I

text: ∂XI

text: n

text: S

text: k

text: Σ6

text: n

text: XIi,

text: XIj

text: X¯Ii,

text: X¯Ij

text: SUGRA

text: ∂yI

text: On

text: the

text: other

text: hand,

text: results

text: in

text: equation

text: show

text: that

text: XIi,

text: XIj

text: X¯Ii,

text: X¯Ij

text: Thus,

text: we

text: can

text: conclude

text: that

text: the

text: action

text: given

text: above

text: tends

text: to

text: an

text: action

text: on

text: the

text: three-dimensional

text: manifold.

text: XIi,

text: XIj

text: X¯Ii,

text: X¯Ij

text: S

text: k

text: SUGRA

text: k

text: k

text: where

text: R

text: is

text: the

text: radius

text: of

text: manifold.

text: By

text: redeﬁning

text: scalar

text: strings,

text: we

text: are

text: able

text: to

text: show

text: that

text: the

text: action

text: of

text: supergravity

text: shrinks

text: to

text: one.

text: X

text: (k

text: X¯

text: (k

text: S

text: (k

text: (k

text: SUGRA

text: This

text: equation

text: yields

text: results

text: that

text: deserves

text: our

text: comments.

text: In

text: fact,

text: two

text: eleven-dimensional

text: manifolds

text: and

text: the

text: bulk

text: interact

text: with

text: each

text: other

text: via

text: diﬀerent

text: types

text: of

text: C,

text: G

text: and

text: Chern-Simons-ﬁelds.

text: When

text: we

text: sum

text: over

text: supergravities

text: that

text: live

text: on

text: these

text: manifolds

text: and

text: consider

text: ﬁelds

text: in

text: the

text: space

text: between

text: them,

text: we

text: get

text: supergravity

text: in

text: space.

text: By

text: canceling

text: the

text: anomaly

text: in

text: this

text: new

text: supergravity,

text: we

text: can

text: obtain

text: the

text: relations

text: between

text: ﬁelds.

text: By

text: substituting

text: these

text: relations

text: into

text: the

text: action

text: of

text: the

text: supergravity,

text: we

text: simply

text: obtain

text: one.

text: This

text: means

text: that

text: the

text: manifold

text: with

text: all

text: its

text: matter

text: content

text: can

text: be

text: topologically

text: shrunk

text: to

text: one

text: point

text: (See

text: In

text: fact,

text: the

text: system

text: of

text: the

text: world

text: began

text: from

text: this

text: point

text: and

text: then

text: expand

text: and

text: construct

text: a

text: world

text: similar

text: to

text: what

text: happens

text: in

text: a

text: Big

text: Bang

text: theory.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 FIG. 1: Symmetry breaking (phase transition) patterns for the non-Abelian gauge group extended models with the evolution of our universe, where the ﬁrst-order phase transitions may happen. In Section III, we will study the phase transition GWs spectra in some concrete hidden gauge group extended models. In Section IV, we show our ﬁnal discussions and conclusions. II. FIRST-ORDER PHASE TRANSITION GWS SPECTRUM In a large class of new physics models with extended non-Abelian hidden gauge group, one or several times cosmological phase transitions can occur during each step’s hidden gauge symmetry breaking. In general, we can have the gauge symmetry breaking patterns with the extended hidden gauge group at each scales with the evolution of the universe as shown in Fig. 1. For example, the hidden symmetry breaking pattern may be the following: · · · → → ⊗ ⊗ → ⊗ ⊗ → ⊗ G(Hidden N) G(Hidden1) G(SU(3) SU(3) U(1) ) G(SU(3) SU(2) U(1) ) G(SU(3) U(1) ). C L Y C L X C EM With the evolution of our universe, symmetry breaking will happen at corresponding energy scale, where the strong ﬁrst-order phase transition may take place. Detailed models are given in Section III. Along with the symmetry breaking from a ‘false’ vacuum to a ‘true’ vacuum, strong ﬁrst-order phase transition occurs if there exists a suﬃcient potential barrier between them. This process can produce an observable stochastic GWs signals, which can be detected in some GWs detectors, such as aLIGO, eLISA, BBO, SKA, FAST and so on. Their sensitivity range for some critical temperatures depends on the energy scale of the ﬁrst-order phase transition for diﬀerent hidden gauge group extended models, as shown in section III. To discuss the GWs spectrum from the ﬁrst-order phase transition, it is necessary to begin with the one-loop ﬁnite temperature eﬀective potential: V (Φ, T) = V (Φ) + V (Φ) + V (Φ, T) + V (Φ, T), (1) eﬀ tree CW thermal daisy where Φ represents the order parameter of the phase transition (a real scalar ﬁeld), V (Φ) is the one-loop CW Coleman-Weinberg potential at T = 0, and V (Φ, T) + V (Φ, T) is the thermal contribution including the thermal daisy daisy resummation [28]. During each step of symmetry breaking in the hidden gauge group extended models, strong ﬁrst-order phase transition may occur. During the ﬁrst-order phase transition, bubbles are nucleated via quantum tunneling or thermally − ﬂuctua∝ting the potential barrier. The bubble nucleation rate per unit volume Γ is given by Γ = Γ0(T)e SE(T) with Γ0(T) T4 [29], where SE(T) is the Eu(cid:90)clidean ac(cid:34)tio(cid:18)n [30(cid:19), 31] deﬁned as (cid:35) 1 dΦ 2 1 ∇ S (T) = dτd3x + ( Φ)2 + V (Φ, T) . (2) E 2 dτ 2 eﬀ (cid:39) − Here, SE(T) S3(T)/T, and Γ = Γ0e S3/T [29] where ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Symmetry

text: breaking

text: (phase

text: transition)

text: patterns

text: for

text: the

text: non-Abelian

text: gauge

text: group

text: extended

text: models

text: with

text: the

text: evolution

text: of

text: our

text: universe,

text: where

text: the

text: ﬁrst-order

text: phase

text: transitions

text: may

text: happen.

text: In

text: Section

text: III,

text: we

text: will

text: study

text: the

text: phase

text: transition

text: GWs

text: spectra

text: in

text: some

text: concrete

text: hidden

text: gauge

text: group

text: extended

text: models.

text: In

text: Section

text: IV,

text: we

text: show

text: our

text: ﬁnal

text: discussions

text: and

text: conclusions.

text: II.

label: section

text: FIRST-ORDER

text: PHASE

text: TRANSITION

text: GWS

text: SPECTRUM

text: In

text: a

text: large

text: class

text: of

text: new

text: physics

text: models

text: with

text: extended

text: non-Abelian

text: hidden

text: gauge

text: group,

text: one

text: or

text: several

text: times

text: cosmological

text: phase

text: transitions

text: can

text: occur

text: during

text: each

text: step’s

text: hidden

text: gauge

text: symmetry

text: breaking.

text: In

text: general,

text: we

text: can

text: have

text: the

text: gauge

text: symmetry

text: breaking

text: patterns

text: with

text: the

text: extended

text: hidden

text: gauge

text: group

text: at

text: each

text: scales

text: with

text: the

text: evolution

text: of

text: the

text: universe

text: as

text: shown

text: in

text: Fig.

text: For

text: example,

text: the

text: hidden

text: symmetry

text: breaking

text: pattern

text: may

text: be

text: the

text: following:

text: G(Hidden

text: N)

text: C

text: L

text: Y

text: C

text: L

text: X

text: C

text: EM

text: With

text: the

text: evolution

text: of

text: our

text: universe,

text: symmetry

text: breaking

text: will

text: happen

text: at

text: corresponding

text: energy

text: scale,

text: where

text: the

text: strong

text: ﬁrst-order

text: phase

text: transition

text: may

text: take

text: place.

text: Detailed

text: models

text: are

text: given

text: in

text: Section

text: III.

text: Along

text: with

text: the

text: symmetry

text: breaking

text: from

text: a

text: ‘false’

text: vacuum

text: to

text: a

text: ‘true’

text: vacuum,

text: strong

text: ﬁrst-order

text: phase

text: transition

text: occurs

text: if

text: there

text: exists

text: a

text: suﬃcient

text: potential

text: barrier

text: between

text: them.

text: This

text: process

text: can

text: produce

text: an

text: observable

text: stochastic

text: GWs

text: signals,

text: which

text: can

text: be

text: detected

text: in

text: some

text: GWs

text: detectors,

text: such

text: as

text: aLIGO,

text: eLISA,

text: BBO,

text: SKA,

text: FAST

text: and

text: so

text: on.

text: Their

text: sensitivity

text: range

text: for

text: some

text: critical

text: temperatures

text: depends

text: on

text: the

text: energy

text: scale

text: of

text: the

text: ﬁrst-order

text: phase

text: transition

text: for

text: diﬀerent

text: hidden

text: gauge

text: group

text: extended

text: models,

text: as

text: shown

text: in

text: section

text: III.

text: To

text: discuss

text: the

text: GWs

text: spectrum

text: from

text: the

text: ﬁrst-order

text: phase

text: transition,

text: it

text: is

text: necessary

text: to

text: begin

text: with

text: the

text: one-loop

text: ﬁnite

text: temperature

text: eﬀective

text: potential:

text: V

text: (Φ,

text: T)

text: V

text: (Φ)

text: V

text: (Φ)

text: V

text: (Φ,

text: T)

text: V

text: (Φ,

text: T),

text: eﬀ

text: tree

text: CW

text: thermal

text: daisy

text: where

text: Φ

text: represents

text: the

text: order

text: parameter

text: of

text: the

text: phase

text: transition

text: (a

text: real

text: scalar

text: ﬁeld),

text: V

text: (Φ)

text: is

text: the

text: one-loop

text: CW

text: Coleman-Weinberg

text: potential

text: at

text: T

text: and

text: V

text: (Φ,

text: T)

text: V

text: (Φ,

text: T)

text: is

text: the

text: thermal

text: contribution

text: including

text: the

text: thermal

text: daisy

text: daisy

text: resummation

text: During

text: each

text: step

text: of

text: symmetry

text: breaking

text: in

text: the

text: hidden

text: gauge

text: group

text: extended

text: models,

text: strong

text: ﬁrst-order

text: phase

text: transition

text: may

text: occur.

text: During

text: the

text: ﬁrst-order

text: phase

text: transition,

text: bubbles

text: are

text: nucleated

text: via

text: quantum

text: tunneling

text: or

text: thermally

text: ﬂuctua∝ting

text: the

text: potential

text: barrier.

text: The

text: bubble

text: nucleation

text: rate

text: per

text: unit

text: volume

text: Γ

text: is

text: given

text: by

text: Γ

text: SE(T)

text: with

text: where

text: SE(T)

text: is

text: the

text: deﬁned

text: as

text: dΦ

text: S

text: (T)

text: V

text: (Φ,

text: T)

text: E

text: dτ

text: eﬀ

text: Here,

text: SE(T)

text: and

text: Γ

text: where

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 (cid:90) (cid:20) (cid:21) 1 ∇ S (T) = d3x ( Φ)2 + V (Φ, T) . (3) 3 2 eﬀ From the above equations, in order to obtain the bubble nucleation rate, the proﬁle of the scalar ﬁeld Φ during the bubble nucleation needs to be calculated by solving the following bounce equation: d2Φ 2 dΦ − ∂V (Φ, T) + eﬀ = 0, (4) dr2 r dr ∂Φ with the boundary conditions dΦ (r = 0) = 0, (5) dr ∞ Φ(r = ) = Φ . (6) false The bounce equation can be solved numerically using the overshoot/undershoot method. The ﬁrst-order phase O (cid:39) transition terminates when nucleation probability of one bubble per horizon volume is of (1), i.e., Γ(T∗) H∗4. That is to say, it should satisfy S (T∗)/T∗ = 4 ln(T∗/100GeV) + 137. (7) 3 It is known that there exist three major sources for producing GWs during the ﬁrst-order phase transition, which are collisions of the vacuum bubbles [15], sound waves [16] and turbulence [17, 18] in the plasma after collisions, respectively. The most well-known source is the bubbles collisions, and the ﬁrst-order phase transition GWs spectrum depends on four parameters. The ﬁrst one is the ratio α of the vacuum energy density released in the phase transition to that of the thermal bath, deﬁned as ≡ (cid:15)(T∗) α , (8) ρ (T∗) rad ∗ where speciﬁes that the quantity is evaluated at the time T∗ determined by Eq.(7). Note that, the false vacuum − | energy density (cid:15)(T∗) = [T dVdeTmffin Vemﬀin(T)] T=T∗ is the latent heat density, and the plasma thermal energy density ρ (T∗) is equal to π2 g∗(T)T4. Here, V min(T) is the temperature-dependent true minimum of the eﬀective potential rad 30 eff of the scalar ﬁeld, which is responsible for the phase transition. The parameter α measures the strength of the phase transition GWs, namely, larger values for α correspond to stronger ﬁrst-order phase transition GWs. − ≡ − | (cid:39) | The se−cond one is the time duration of the phase transition β 1, where one has β ddStE t=t∗ Γ1 ddΓt t=t∗. In other words, β 1 corresponds to the typical time scale of the phase transition. Since β = Γ˙ /Γ during the phase transition from its deﬁnition, one has (cid:12) (cid:12) (cid:12) (cid:12) β d(S /T) = T 3 . (9) H∗ dT T=T∗ The third one is the eﬃciency factor λ, which characterizes the fraction of the energy density converted into the motion of the colliding bubble walls and the last one is the bubble wall velocity v . The energy released into the GWs b of peak frequency is then given by [32]: (cid:18) (cid:19) ρ ∼ H∗ 2 α2 GW θ λ2 v3. (10) ρ β (1 + α)2 b tot ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: (T)

text: V

text: (Φ,

text: T)

text: eﬀ

text: From

text: the

text: above

text: equations,

text: in

text: order

text: to

text: obtain

text: the

text: bubble

text: nucleation

text: rate,

text: the

text: proﬁle

text: of

text: the

text: scalar

text: ﬁeld

text: Φ

text: during

text: the

text: bubble

text: nucleation

text: needs

text: to

text: be

text: calculated

text: by

text: solving

text: the

text: following

text: bounce

text: equation:

text: dΦ

text: ∂V

text: (Φ,

text: T)

text: eﬀ

text: r

text: dr

text: ∂Φ

text: with

text: the

text: boundary

text: conditions

text: dΦ

text: (r

text: dr

text: Φ(r

text: Φ

text: false

text: The

text: bounce

text: equation

text: can

text: be

text: solved

text: numerically

text: using

text: the

text: overshoot/undershoot

text: method.

text: The

text: ﬁrst-order

text: phase

text: O

text: transition

text: terminates

text: when

text: nucleation

text: probability

text: of

text: one

text: bubble

text: per

text: horizon

text: volume

text: is

text: of

text: i.e.,

text: Γ(T∗)

text: That

text: is

text: to

text: say,

text: it

text: should

text: satisfy

text: S

text: (T∗)/T∗

text: It

text: is

text: known

text: that

text: there

text: exist

text: three

text: major

text: sources

text: for

text: producing

text: GWs

text: during

text: the

text: ﬁrst-order

text: phase

text: transition,

text: which

text: are

text: collisions

text: of

text: the

text: vacuum

text: bubbles

text: sound

text: waves

text: and

text: turbulence

text: in

text: the

text: plasma

text: after

text: collisions,

text: respectively.

text: The

text: most

text: well-known

text: source

text: is

text: the

text: bubbles

text: collisions,

text: and

text: the

text: ﬁrst-order

text: phase

text: transition

text: GWs

text: spectrum

text: depends

text: on

text: four

text: parameters.

text: The

text: ﬁrst

text: one

text: is

text: the

text: ratio

text: α

text: of

text: the

text: vacuum

text: energy

text: density

text: released

text: in

text: the

text: phase

text: transition

text: to

text: that

text: of

text: the

text: thermal

text: bath,

text: deﬁned

text: as

text: α

text: ρ

text: (T∗)

text: rad

text: where

text: speciﬁes

text: that

text: the

text: quantity

text: is

text: evaluated

text: at

text: the

text: time

text: T∗

text: determined

text: by

text: Note

text: that,

text: the

text: false

text: vacuum

text: energy

text: density

text: [T

text: dVdeTmffin

text: Vemﬀin(T)]

text: T=T∗

text: is

text: the

text: latent

text: heat

text: density,

text: and

text: the

text: plasma

text: thermal

text: energy

text: density

text: ρ

text: (T∗)

text: is

text: equal

text: to

text: π2

text: Here,

text: V

text: min(T)

text: is

text: the

text: temperature-dependent

text: true

text: minimum

text: of

text: the

text: eﬀective

text: potential

text: rad

text: eff

text: of

text: the

text: scalar

text: ﬁeld,

text: which

text: is

text: responsible

text: for

text: the

text: phase

text: transition.

text: The

text: parameter

text: α

text: measures

text: the

text: strength

text: of

text: the

text: phase

text: transition

text: GWs,

text: namely,

text: larger

text: values

text: for

text: α

text: correspond

text: to

text: stronger

text: ﬁrst-order

text: phase

text: transition

text: GWs.

text: The

text: se−cond

text: one

text: is

text: the

text: time

text: duration

text: of

text: the

text: phase

text: transition

text: β

text: where

text: one

text: has

text: β

text: ddStE

text: t=t∗

text: Γ1

text: ddΓt

text: t=t∗.

text: In

text: other

text: words,

text: β

text: corresponds

text: to

text: the

text: typical

text: time

text: scale

text: of

text: the

text: phase

text: transition.

text: Since

text: β

text: Γ˙

text: /Γ

text: during

text: the

text: phase

text: transition

text: from

text: its

text: deﬁnition,

text: one

text: has

text: β

text: d(S

text: /T)

text: T

text: H∗

text: dT

text: T=T∗

text: The

text: third

text: one

text: is

text: the

text: eﬃciency

text: factor

text: λ,

text: which

text: characterizes

text: the

text: fraction

text: of

text: the

text: energy

text: density

text: converted

text: into

text: the

text: motion

text: of

text: the

text: colliding

text: bubble

text: walls

text: and

text: the

text: last

text: one

text: is

text: the

text: bubble

text: wall

text: velocity

text: v

text: The

text: energy

text: released

text: into

text: the

text: GWs

text: b

text: of

text: peak

text: frequency

text: is

text: then

text: given

text: by

text: ρ

text: H∗

text: α2

text: GW

text: θ

text: λ2

text: ρ

text: β

text: b

text: tot

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 III. PHASE TRANSITION GWS FROM NON-ABELIAN GAUGE GROUP EXTENDED MODELS In this section, we discuss the phase transition GWs in some non-Abelian gauge group extended models in details, where one or several times symmetry breaking and ﬁrst-order phase transition may occur with the evolution of our universe at certain critical temperature. Firstly, the GWs spectra in the hidden gauge group extended models based ⊗ ⊗ on the SU(3) SU(3) U(1) gauge symmetry, commonly known as the 3-3-1 model [20, 21] are investigated. c L Y The 3-3-1 models can explain the electric charge quantization and three generations of fermions [20, 21]. The collider phenomenology of the 3-3-1 models has been extensively studied, such as the recent Ref. [38] and references therein, and the phase transitions in some versions of 3-3-1 models have been studied by Refs. [39–41]. Here, we try to explore ⊗ → ⊗ → the possible phase transition patterns associated with the symmetry breaking SU(3) U(1) SU(2) U(1) L Y L X U(1) using the phase transition GWs in some versions of the 3-3-1 models, where the scalars ﬁelds are accommodated EM in a certain representation of the SU(3) gauge group in each version. L In the following, we will show three versions of 3-3-1 models (the minimal, the economical and the reduced minimal 3-3-1 models, respectively) [39–41], where very strong ﬁrst-order phase transitions can take place at the TeV scale [39– 41] and produce detectable phase transition GWs by eLISA and BBO. For simplicity, we limit our discussion of the (cid:0) (cid:1) ﬁrst-order phase transition to the thermal barrier case, where the potential barrier in the ﬁnite temperature eﬀective potential origins from thermal loop eﬀects. In this type of thermal barrier case, the bosonic ﬁelds contribute to the (cid:51) − thermal eﬀective potential of the form V ( T/12π) m2 (X, T) 3/2 in the limit of high-temperature expansion. eﬀ boson To begin with the concrete prediction of the GWs in the following examples, we ﬁrst discuss the general eﬀective potential near the symmetry breaking and phase transition temperature, which can be approximated by (cid:0) (cid:1) ≈ 1 − − e T λ V (X, T) µ2 + c T2 X2 (X2)3/2 + X4. (14) eﬀ 2 12π 4 (cid:80) Here, X represents the ﬁeld of the order parameter for the phase transition. For the electroweak phase transition in the SM, X ﬁeld is just the Higgs ﬁeld. The parameter e quantify the interactions between X ﬁeld and the light ∼ × bosons, and can be schematically written as e (degrees of freedom) (coupling to X)3/2. And, the light bosonic ﬁelds parameters c depends on interaction between X and light bosons and fermions. For the heavy ﬁelds whose masses are much larger than the critical temperature, their contribution can be omitted from Boltzmann suppression. This can help to simplify our discussions when the models have many new ﬁelds at diﬀerent energy scales. Thus, in this case of qualitative analysis, the wash out parameter can be obtained as (cid:104) (cid:105) X (T ) ≈ e c , (15) T 6πλ c where the angle bracket means the vacuum expectation value (VEV) of the ﬁeld X at the critical temperature T . c From the above qualitative analysis, we know that introducing new light bosonic ﬁelds (compared to the corresponding critical temperature) helps to produce or enhance the ﬁrst-order phase transition. The 3-3-1 models just introduce enough bosonic ﬁelds to produce detectable phase transition GWs. A. Phase transition GWs spectrum in the economical 3-3-1 model and the reduced minimal 3-3-1 model We consider the ﬁrst-order phase transition GWs spectrum in the so-called economical 3-3-1 model [40]. In this version of 3-3-1 model, one chooses the simplest SU(3) representations for the scalar ﬁelds with spontaneously L symmetry breaking, namely, two complex scalar triplets with di(cid:18)ﬀerent h(cid:19)ypercharge are needed as follows (cid:0) (cid:1) − ∼ −1 χ = χ0, χ , χ0 T 3, , (16) 1 2 3 3 (cid:18) (cid:19) (cid:0) (cid:1) ∼ 2 φ = φ+, φ0, φ+ T 3, . (17) 1 2 3 3 The scalar potential is written as ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: III.

text: PHASE

text: TRANSITION

text: GWS

text: FROM

text: NON-ABELIAN

text: GAUGE

text: GROUP

text: EXTENDED

text: MODELS

text: In

text: this

text: section,

text: we

text: discuss

text: the

text: phase

text: transition

text: GWs

text: in

text: some

text: non-Abelian

text: gauge

text: group

text: extended

text: models

text: in

text: details,

text: where

text: one

text: or

text: several

text: times

text: symmetry

text: breaking

text: and

text: ﬁrst-order

text: phase

text: transition

text: may

text: occur

text: with

text: the

text: evolution

text: of

text: our

text: universe

text: at

text: certain

text: critical

text: temperature.

text: Firstly,

text: the

text: GWs

text: spectra

text: in

text: the

text: hidden

text: gauge

text: group

text: extended

text: models

text: based

text: on

text: the

text: gauge

text: symmetry,

text: commonly

text: known

text: as

text: the

text: model

text: are

text: investigated.

text: c

text: L

text: Y

text: The

text: models

text: can

text: explain

text: the

text: electric

text: charge

text: quantization

text: and

text: three

text: generations

text: of

text: fermions

text: The

text: collider

text: phenomenology

text: of

text: the

text: models

text: has

text: been

text: extensively

text: studied,

text: such

text: as

text: the

text: recent

text: Ref.

text: and

text: references

text: therein,

text: and

text: the

text: phase

text: transitions

text: in

text: some

text: versions

text: of

text: models

text: have

text: been

text: studied

text: by

text: Refs.

text: Here,

text: we

text: try

text: to

text: explore

text: the

text: possible

text: phase

text: transition

text: patterns

text: associated

text: with

text: the

text: symmetry

text: breaking

text: L

text: Y

text: L

text: X

text: using

text: the

text: phase

text: transition

text: GWs

text: in

text: some

text: versions

text: of

text: the

text: models,

text: where

text: the

text: scalars

text: ﬁelds

text: are

text: accommodated

text: EM

text: in

text: a

text: certain

text: representation

text: of

text: the

text: gauge

text: group

text: in

text: each

text: version.

text: L

text: In

text: the

text: following,

text: we

text: will

text: show

text: three

text: versions

text: of

text: models

text: (the

text: minimal,

text: the

text: economical

text: and

text: the

text: reduced

text: minimal

text: models,

text: respectively)

text: where

text: very

text: strong

text: ﬁrst-order

text: phase

text: transitions

text: can

text: take

text: place

text: at

text: the

text: TeV

text: scale

text: and

text: produce

text: detectable

text: phase

text: transition

text: GWs

text: by

text: eLISA

text: and

text: BBO.

text: For

text: simplicity,

text: we

text: limit

text: our

text: discussion

text: of

text: the

text: ﬁrst-order

text: phase

text: transition

text: to

text: the

text: thermal

text: barrier

text: case,

text: where

text: the

text: potential

text: barrier

text: in

text: the

text: ﬁnite

text: temperature

text: eﬀective

text: potential

text: origins

text: from

text: thermal

text: loop

text: eﬀects.

text: In

text: this

text: type

text: of

text: thermal

text: barrier

text: case,

text: the

text: bosonic

text: ﬁelds

text: contribute

text: to

text: the

text: thermal

text: eﬀective

text: potential

text: of

text: the

text: form

text: V

text: (X,

text: T)

text: in

text: the

text: limit

text: of

text: high-temperature

text: expansion.

text: eﬀ

text: boson

text: To

text: begin

text: with

text: the

text: concrete

text: prediction

text: of

text: the

text: GWs

text: in

text: the

text: following

text: examples,

text: we

text: ﬁrst

text: discuss

text: the

text: general

text: eﬀective

text: potential

text: near

text: the

text: symmetry

text: breaking

text: and

text: phase

text: transition

text: temperature,

text: which

text: can

text: be

text: approximated

text: by

text: e

text: T

text: λ

text: V

text: (X,

text: T)

text: µ2

text: c

text: eﬀ

text: Here,

text: X

text: represents

text: the

text: ﬁeld

text: of

text: the

text: order

text: parameter

text: for

text: the

text: phase

text: transition.

text: For

text: the

text: electroweak

text: phase

text: transition

text: in

text: the

text: SM,

text: X

text: ﬁeld

text: is

text: just

text: the

text: Higgs

text: ﬁeld.

text: The

text: parameter

text: e

text: quantify

text: the

text: interactions

text: between

text: X

text: ﬁeld

text: and

text: the

text: light

text: bosons,

text: and

text: can

text: be

text: schematically

text: written

text: as

text: e

text: (degrees

text: of

text: freedom)

text: (coupling

text: to

text: And,

text: the

text: light

text: bosonic

text: ﬁelds

text: parameters

text: c

text: depends

text: on

text: interaction

text: between

text: X

text: and

text: light

text: bosons

text: and

text: fermions.

text: For

text: the

text: heavy

text: ﬁelds

text: whose

text: masses

text: are

text: much

text: larger

text: than

text: the

text: critical

text: temperature,

text: their

text: contribution

text: can

text: be

text: omitted

text: from

text: Boltzmann

text: suppression.

text: This

text: can

text: help

text: to

text: simplify

text: our

text: discussions

text: when

text: the

text: models

text: have

text: many

text: new

text: ﬁelds

text: at

text: diﬀerent

text: energy

text: scales.

text: Thus,

text: in

text: this

text: case

text: of

text: qualitative

text: analysis,

text: the

text: wash

text: out

text: parameter

text: can

text: be

text: obtained

text: as

text: X

text: (T

text: e

text: c

text: T

text: 6πλ

text: c

text: where

text: the

text: angle

text: bracket

text: means

text: the

text: vacuum

text: expectation

text: value

text: (VEV)

text: of

text: the

text: ﬁeld

text: X

text: at

text: the

text: critical

text: temperature

text: T

text: c

text: From

text: the

text: above

text: qualitative

text: analysis,

text: we

text: know

text: that

text: introducing

text: new

text: light

text: bosonic

text: ﬁelds

text: (compared

text: to

text: the

text: corresponding

text: critical

text: temperature)

text: helps

text: to

text: produce

text: or

text: enhance

text: the

text: ﬁrst-order

text: phase

text: transition.

text: The

text: models

text: just

text: introduce

text: enough

text: bosonic

text: ﬁelds

text: to

text: produce

text: detectable

text: phase

text: transition

text: GWs.

text: A.

text: Phase

text: transition

text: GWs

text: spectrum

text: in

text: the

text: economical

text: model

text: and

text: the

text: reduced

text: minimal

text: model

text: We

text: consider

text: the

text: ﬁrst-order

text: phase

text: transition

text: GWs

text: spectrum

text: in

text: the

text: so-called

text: economical

text: model

text: In

text: this

text: version

text: of

text: model,

text: one

text: chooses

text: the

text: simplest

text: representations

text: for

text: the

text: scalar

text: ﬁelds

text: with

text: spontaneously

text: L

text: symmetry

text: breaking,

text: namely,

text: two

text: complex

text: scalar

text: triplets

text: with

text: are

text: needed

text: as

text: follows

text: −1

text: χ

text: χ

text: χ0

text: T

text: φ

text: φ+,

text: φ+

text: T

text: The

text: scalar

text: potential

text: is

text: written

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 hadron (photon - hadron process) [15]. In the partic- hadronic collisions, the diﬀerential cross section can be ular case of exclusive vector meson photoproduction in expressed as follows (cid:20) (cid:21) (cid:20) (cid:21) → ⊗ ⊗ dσ [h + h h V h ] dN | dσ → ⊗ dN | dσ → ⊗ 1 2 1 2 = ω (γh V h ) + ω (γh V h ) (1) dY dt dω h1 dt 2 2 dω h2 dt 1 1 ωL ωR where the rapidity (Y ) of the vector meson in the ﬁ- scribed by the Dress - Zeppenfeld [16] and the relativistic nal state is determined by the photon energy ω in the point – like charge [15] models, respectively. → collider∝frame and by the mass MV of the vector me- In the color dipole formalism, the γh V h process son [Y ln (ω/MV )]. M→oreov⊗er, dσ/dt is the diﬀerential can be factorized in terms of the ﬂuctuation of the virtual cros⊗s section for the γhi V hi process, with the sym- photon into a qq¯ color dipole, the dipole-hadron scatter- bol representing∝th−e presence of ∝a rapidity gap in the ﬁ- ing by a color singlet exchange and the recombination nal state and ωL ( e Y ) and ωR ( eY ) denoting photon into the vector meson V . The ﬁnal state is characterized energies from the h1 and h2 hadrons, respectively. Fur- by the presence of a rapidity gap. The diﬀerential cross thermore, dN denotes the equivalent photon spectrum of section for the exclusive vector meson photoproduction dω the relativistic incident hadron, with the ﬂux of a nucleus can be expressed as follows being enhanced by a factor Z2 in comparison to the pro- ton one. Eq. (1) takes into account the fact that both incident hadrons can be sources of the photons which will dσ 1 |A → | interact with the other hadron, with the ﬁrst term on the = γh V h(x, ∆) 2 , (2) right-hand side of the Eq. (1) being dominant at positive dt 16π rapidities while the second term dominating at negative rapidities due to the fact that the photon ﬂux has support at small values of ω, decreasing exponentially at large ω. with the amplitude for producing an exclusive vector As in Refs. [6, 11] we will assume that the photon ﬂux meson diﬀractively being given in the color dipole for- associated to the proton and to the nucleus can be de- malism by (cid:90) Aγh→V h(x, ∆) = i dz d2r d2bh e−i[bh−(1−z)r].∆ (ΨV ∗Ψ) 2Nh(x, r, bh) (3) where (ΨV ∗Ψ) denotes the wave function overlap betwe√en and thus about the non-linear and quantum eﬀects in the − √ the photon and vector meson wave functions, ∆ = t hadron wave function. It depends on the γh center - of - is the momentum transfer and b is the impact param- mass reaction energy, W = [2ω s]1/2, through the vari- h eter of the dipole relative to the hadron target. More- able x = M2 /W2. As in Refs. [6, 11], in what follows we V over, the variables r and z are the dipole transverse ra- will consider the Boosted Gaussian model [17, 18] for the dius and the momentum fraction of the photon carried overlap function and the impact parameter Color Glass − by a quark (an antiquark carries then 1 z), respectively. Condensate (bCGC) model [18] for the dipole – proton N N h(x, r, b ) is the forward dipole-target scattering am- scattering amplitude p. In this model the dipole - pro- h plitude (for a dipole at impact parameter b ) which en- ton scattering amplitude is given by [18] h codes all the information about the hadronic scattering,  (cid:16) (cid:17)  (cid:16) (cid:17)  Np(x, r, bp) = N−0 −r Qs2(bp) 2 γs+ln(2/κrλQYs(bp)) rQs(bp) ≤ 2 (4) 1 e A ln2 (B r Qs(bp)) rQs(bp) > 2 (cid:48)(cid:48) (cid:48) with κ = χ (γ )/χ (γ ), where χ is the LO BFKL char- acteristic function. The coeﬃcients A and B are deter- s s ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: hadron

text: (photon

text: hadron

text: process)

text: In

text: the

text: partic-

text: hadronic

text: collisions,

text: the

text: diﬀerential

text: cross

text: section

text: can

text: be

text: ular

text: case

text: of

text: exclusive

text: vector

text: meson

text: photoproduction

text: in

text: expressed

text: as

text: follows

text: dσ

text: [h

text: h

text: h

text: V

text: h

text: dN

text: dσ

text: dN

text: dσ

text: ω

text: (γh

text: V

text: h

text: ω

text: (γh

text: V

text: h

text: dY

text: dt

text: dω

text: dt

text: dω

text: dt

text: ωL

text: ωR

text: where

text: the

text: rapidity

text: (Y

text: of

text: the

text: vector

text: meson

text: in

text: the

text: ﬁ-

text: scribed

text: by

text: the

text: Dress

text: Zeppenfeld

text: and

text: the

text: relativistic

text: nal

text: state

text: is

text: determined

text: by

text: the

text: photon

text: energy

text: ω

text: in

text: the

text: point

text: like

text: charge

text: models,

text: respectively.

text: collider∝frame

text: and

text: by

text: the

text: mass

text: MV

text: of

text: the

text: vector

text: me-

text: In

text: the

text: color

text: dipole

text: formalism,

text: the

text: γh

text: V

text: h

text: process

text: son

text: [Y

text: ln

text: (ω/MV

text: M→oreov⊗er,

text: dσ/dt

text: is

text: the

text: diﬀerential

text: can

text: be

text: factorized

text: in

text: terms

text: of

text: the

text: ﬂuctuation

text: of

text: the

text: virtual

text: cros⊗s

text: section

text: for

text: the

text: γhi

text: V

text: hi

text: process,

text: with

text: the

text: sym-

text: photon

text: into

text: a

text: qq¯

text: color

text: dipole,

text: the

text: dipole-hadron

text: scatter-

text: bol

text: representing∝th−e

text: presence

text: of

text: ∝a

text: rapidity

text: gap

text: in

text: the

text: ﬁ-

text: ing

text: by

text: a

text: color

text: singlet

text: exchange

text: and

text: the

text: recombination

text: nal

text: state

text: and

text: ωL

text: e

text: Y

text: and

text: ωR

text: eY

text: denoting

text: photon

text: into

text: the

text: vector

text: meson

text: V

text: The

text: ﬁnal

text: state

text: is

text: characterized

text: energies

text: from

text: the

text: and

text: hadrons,

text: respectively.

text: Fur-

text: by

text: the

text: presence

text: of

text: a

text: rapidity

text: gap.

text: The

text: diﬀerential

text: cross

text: thermore,

text: dN

text: denotes

text: the

text: equivalent

text: photon

text: spectrum

text: of

text: section

text: for

text: the

text: exclusive

text: vector

text: meson

text: photoproduction

text: dω

text: the

text: relativistic

text: incident

text: hadron,

text: with

text: the

text: ﬂux

text: of

text: a

text: nucleus

text: can

text: be

text: expressed

text: as

text: follows

text: being

text: enhanced

text: by

text: a

text: factor

text: in

text: comparison

text: to

text: the

text: pro-

text: ton

text: one.

text: Eq.

text: takes

text: into

text: account

text: the

text: fact

text: that

text: both

text: incident

text: hadrons

text: can

text: be

text: sources

text: of

text: the

text: photons

text: which

text: will

text: dσ

text: |A

text: interact

text: with

text: the

text: other

text: hadron,

text: with

text: the

text: ﬁrst

text: term

text: on

text: the

text: γh

text: V

text: h(x,

text: right-hand

text: side

text: of

text: the

text: Eq.

text: being

text: dominant

text: at

text: positive

text: dt

text: rapidities

text: while

text: the

text: second

text: term

text: dominating

text: at

text: negative

text: rapidities

text: due

text: to

text: the

text: fact

text: that

text: the

text: photon

text: ﬂux

text: has

text: support

text: at

text: small

text: values

text: of

text: ω,

text: decreasing

text: exponentially

text: at

text: large

text: ω.

text: with

text: the

text: amplitude

text: for

text: producing

text: an

text: exclusive

text: vector

text: As

text: in

text: Refs.

text: we

text: will

text: assume

text: that

text: the

text: photon

text: ﬂux

text: meson

text: diﬀractively

text: being

text: given

text: in

text: the

text: color

text: dipole

text: for-

text: associated

text: to

text: the

text: proton

text: and

text: to

text: the

text: nucleus

text: can

text: be

text: de-

text: malism

text: by

text: Aγh→V

text: h(x,

text: i

text: dz

text: (ΨV

text: ∗Ψ)

text: r,

text: bh)

text: where

text: (ΨV

text: ∗Ψ)

text: denotes

text: the

text: wave

text: function

text: overlap

text: betwe√en

text: and

text: thus

text: about

text: the

text: non-linear

text: and

text: quantum

text: eﬀects

text: in

text: the

text: the

text: photon

text: and

text: vector

text: meson

text: wave

text: functions,

text: t

text: hadron

text: wave

text: function.

text: It

text: depends

text: on

text: the

text: γh

text: center

text: of

text: is

text: the

text: momentum

text: transfer

text: and

text: b

text: is

text: the

text: impact

text: param-

text: mass

text: reaction

text: energy,

text: W

text: through

text: the

text: vari-

text: h

text: eter

text: of

text: the

text: dipole

text: relative

text: to

text: the

text: hadron

text: target.

text: More-

text: able

text: x

text: As

text: in

text: Refs.

text: in

text: what

text: follows

text: we

text: V

text: over,

text: the

text: variables

text: r

text: and

text: z

text: are

text: the

text: dipole

text: transverse

text: ra-

text: will

text: consider

text: the

text: Boosted

text: Gaussian

text: model

text: for

text: the

text: dius

text: and

text: the

text: momentum

text: fraction

text: of

text: the

text: photon

text: carried

text: overlap

text: function

text: and

text: the

text: impact

text: parameter

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text: 7 experimental LHC data. In order to estimate the impact independently of the fact that this contribution could not of the gluon saturation eﬀects, we also have presented been easily experimentally separated. These results are a comparison with the predictions obtained disregarding robust predictions of the saturation physics, which can these eﬀects. Our results demonstrate that gluon satu- be used to investigate non-linear QCD dynamics in the ration eﬀects reduce the magnitude of the cross sections, kinematical range of the Run 2 of the LHC. with the reduction being larger for lighter vector mesons. Moreover, the gluon saturation eﬀects change the posi- Acknowledgments tions of the dips with respect to the linear regime, shifting the dips to smaller values of the transverse momentum. Finally, our results indicate that dips predicted by the co- This work was partially ﬁnanced by the Brazilian fund- herent contribution in PbPb collisions should be visible, ing agencies CNPq, CAPES, FAPESP and FAPERGS. [1] K. Akiba et al. [LHC Forward Physics Working Group arXiv:1612.06254 [hep-ph]. Collaboration], J. Phys. G 43, 110201 (2016) [12] V. Guzey, M. Strikman and M. Zhalov, arXiv:1611.05471 [2] V. P. Goncalves and C. A. Bertulani, Phys. Rev. C 65, [hep-ph]. 054905 (2002). [13] V. P. Gon¸calves and D. S. Pires, Phys. Rev. C 91, 055207 [3] V. P. Goncalves and M. V. T. Machado, Eur. Phys. J. C (2015) 40, 519 (2005). [14] V. P. Goncalves, F. S. Navarra and D. Spiering, J. Phys. [4] H. Mantysaari and B. Schenke, Phys. Rev. Lett. 117, no. G 43, no. 9, 095002 (2016) 5, 052301 (2016); Phys. Rev. D 94, 034042 (2016) [15] G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Y. [5] V. P. Goncalves and M. V. T. Machado, Phys. Rev. C 73, Kharlov, Phys. Rep. 364, 359 (2002); V. P. Goncalves 044902 (2006); Phys. Rev. D 77, 014037 (2008); Phys. and M. V. T. Machado, Mod. Phys. Lett. A 19, 2525 Rev. C 84, 011902 (2011) (2004); C. A. Bertulani, S. R. Klein and J. Nystrand, [6] V. P. Goncalves, B. D. Moreira and F. S. Navarra, Phys. Ann. Rev. Nucl. Part. Sci. 55, 271 (2005); K. Hencken et Rev. C 90, 015203 (2014); Phys. Lett. B 742, 172 (2015). al., Phys. Rept. 458, 1 (2008). [7] W. Schafer and A. Szczurek, Phys. Rev. D 76, 094014 [16] M. Drees and D. Zeppenfeld, Phys. Rev. D 39, 2536 (2007); A. Rybarska, W. Schafer and A. Szczurek, Phys. (1989). Lett. B 668, 126 (2008); A. Cisek, W. Schafer and [17] H. Kowalski and D. Teaney, Phys. Rev. D 68, 114005 A. Szczurek, Phys. Rev. C 86, 014905 (2012) (2003) [8] L. Frankfurt, V. Guzey, M. Strikman and M. Zhalov, [18] H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D 74, JHEP 0308, 043 (2003); V. Guzey and M. Zhalov, JHEP 074016 (2006) 1310, 207 (2013); JHEP 1402, 046 (2014). [19] A. H. Rezaeian and I. Schmidt, Phys. Rev. D 88, 074016 [9] S. P. Jones, A. D. Martin, M. G. Ryskin and T. Teubner, (2013) JHEP 1311, 085 (2013); Eur. Phys. J. C 76, no. 11, 633 [20] T. Lappi and H. Mantysaari, Phys. Rev. C 83, 065202 (2016) (2011) [10] G. Sampaio dos Santos and M. V. T. Machado, J. [21] T. Toll and T. Ullrich, Phys. Rev. C 87, 024913 (2013) Phys. G 42, 105001 (2015); M. B. G. Ducati, F. Kopp, [22] A. Caldwell and H. Kowalski, Phys. Rev. C 81, 025203 M. V. T. Machado and S. Martins, Phys. Rev. D 94, (2010). 094023 (2016). [11] V. P. Goncalves, B. D. Moreira and F. S. Navarra, ##LTLine## ##LTLine## ##LTLine##

text: experimental

text: LHC

text: data.

text: In

text: order

text: to

text: estimate

text: the

text: impact

text: independently

text: of

text: the

text: fact

text: that

text: this

text: contribution

text: could

text: not

text: of

text: the

text: gluon

text: saturation

text: eﬀects,

text: we

text: also

text: have

text: presented

text: been

text: easily

text: experimentally

text: separated.

text: These

text: results

text: are

text: a

text: comparison

text: with

text: the

text: predictions

text: obtained

text: disregarding

text: robust

text: predictions

text: of

text: the

text: saturation

text: physics,

text: which

text: can

text: these

text: eﬀects.

text: Our

text: results

text: demonstrate

text: that

text: gluon

text: satu-

text: be

text: used

text: to

text: investigate

text: non-linear

text: QCD

text: dynamics

text: in

text: the

text: ration

text: eﬀects

text: reduce

text: the

text: magnitude

text: of

text: the

text: cross

text: sections,

text: kinematical

text: range

text: of

text: the

text: Run

text: of

text: the

text: LHC.

text: with

text: the

text: reduction

text: being

text: larger

text: for

text: lighter

text: vector

text: mesons.

text: Moreover,

text: the

text: gluon

text: saturation

text: eﬀects

text: change

text: the

text: posi-

text: Acknowledgments

text: tions

text: of

text: the

text: dips

text: with

text: respect

text: to

text: the

text: linear

text: regime,

text: shifting

text: the

text: dips

text: to

text: smaller

text: values

text: of

text: the

text: transverse

text: momentum.

text: Finally,

text: our

text: results

text: indicate

text: that

text: dips

text: predicted

text: by

text: the

text: co-

text: This

text: work

text: was

text: partially

text: ﬁnanced

text: by

text: the

text: Brazilian

text: fund-

text: herent

text: contribution

text: in

text: PbPb

text: collisions

text: should

text: be

text: visible,

text: ing

text: agencies

text: CNPq,

text: CAPES,

text: FAPESP

text: and

text: FAPERGS.

text: K.

text: Akiba

text: et

text: al.

text: [LHC

text: Forward

text: Physics

text: Working

text: Group

text: [hep-ph].

text: Collaboration],

text: J.

text: Phys.

text: G

text: V.

text: Guzey,

text: M.

text: Strikman

text: and

text: M.

text: Zhalov,

text: V.

text: P.

text: Goncalves

text: and

text: C.

text: A.

text: Bertulani,

text: Phys.

text: Rev.

text: C

text: [hep-ph].

text: V.

text: P.

text: Gon¸calves

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text: S.

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text: Phys.

text: Rev.

text: C

text: V.

text: P.

text: Goncalves

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text: V.

text: T.

text: Machado,

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text: Phys.

text: J.

text: C

text: V.

text: P.

text: Goncalves,

text: F.

text: S.

text: Navarra

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text: Spiering,

text: J.

text: Phys.

text: H.

text: Mantysaari

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text: Schenke,

text: Phys.

text: Rev.

text: Lett.

text: no.

text: G

text: no.

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text: Rev.

text: D

text: G.

text: Baur,

text: K.

text: Hencken,

text: D.

text: Trautmann,

text: S.

text: Sadovsky,

text: Y.

text: V.

text: P.

text: Goncalves

text: and

text: M.

text: V.

text: T.

text: Machado,

text: Phys.

text: Rev.

text: C

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text: Phys.

text: Rep.

text: V.

text: P.

text: Goncalves

text: Phys.

text: Rev.

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text: Phys.

text: and

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text: V.

text: T.

text: Machado,

text: Mod.

text: Phys.

text: Lett.

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text: Rev.

text: C

text: C.

text: A.

text: Bertulani,

text: S.

text: R.

text: Klein

text: and

text: J.

text: Nystrand,

text: V.

text: P.

text: Goncalves,

text: B.

text: D.

text: Moreira

text: and

text: F.

text: S.

text: Navarra,

text: Phys.

text: Ann.

text: Rev.

text: Nucl.

text: Part.

text: Sci.

text: K.

text: Hencken

text: et

text: Rev.

text: C

text: Phys.

text: Lett.

text: B

text: al.,

text: Phys.

text: Rept.

text: W.

text: Schafer

text: and

text: A.

text: Szczurek,

text: Phys.

text: Rev.

text: D

text: M.

text: Drees

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text: Zeppenfeld,

text: Phys.

text: Rev.

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text: W.

text: Schafer

text: and

text: A.

text: Szczurek,

text: Phys.

text: Lett.

text: B

text: A.

text: Cisek,

text: W.

text: Schafer

text: and

text: H.

text: Kowalski

text: and

text: D.

text: Teaney,

text: Phys.

text: Rev.

text: D

text: A.

text: Szczurek,

text: Phys.

text: Rev.

text: C

text: L.

text: Frankfurt,

text: V.

text: Guzey,

text: M.

text: Strikman

text: and

text: M.

text: Zhalov,

text: H.

text: Kowalski,

text: L.

text: Motyka

text: and

text: G.

text: Watt,

text: Phys.

text: Rev.

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text: V.

text: Guzey

text: and

text: M.

text: Zhalov,

text: JHEP

text: JHEP

text: A.

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text: and

text: I.

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text: Phys.

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text: D

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text: A.

text: D.

text: Martin,

text: M.

text: G.

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text: T.

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text: no.

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text: and

text: H.

text: Mantysaari,

text: Phys.

text: Rev.

text: C

text: G.

text: Sampaio

text: dos

text: Santos

text: and

text: M.

text: V.

text: T.

text: Machado,

text: J.

text: T.

text: Toll

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text: T.

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text: Phys.

text: Rev.

text: C

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text: G

text: M.

text: B.

text: G.

text: Ducati,

text: F.

text: Kopp,

text: A.

text: Caldwell

text: and

text: H.

text: Kowalski,

text: Phys.

text: Rev.

text: C

text: M.

text: V.

text: T.

text: Machado

text: and

text: S.

text: Martins,

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text: Rev.

text: D

text: V.

text: P.

text: Goncalves,

text: B.

text: D.

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text: F.

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text: 6 R. Mojgani and M. Balajewicz 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0 0.5 1 1.5 0 0.5 1 1.5 x x (a) Traditional Eulerian MOR. (b) Lagrangian MOR. Figure 1: Model order reduction of scalar convection equation; HFM (thick grey), k = 2 ROMs (dashed green), and k = 5 ROMs (solid red). Solutions are plotted for t = 0, 1/3, 2/3, 1. 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0 0.5 1 1.5 0 0.5 1 1.5 x x (a) Traditional Eulerian MOR. (b) Lagrangian MOR. Figure 2: Model order reduction of scalar convection-diﬀusion equation with Pe = 103; HFM (thick grey), k = 2 ROMs (dashed green), and k = 5 ROMs (solid red). Solutions are plotted for t = 0, 1/3, 2/3, 1. traditional Eulerian ROMs and the new, Lagrangian ROMs, are illustrated in in the (a) and (b) subﬁgures. Solutions are plotted for t = 0, 1/3, 2/3, 1. In both ﬁgures, thick grey lines correspond to the HFM models, while the dashed green and solid red lines correspond to k = 2, and k = 5 ROMs, respectively. Convergence of Eulerian and Lagrangian ROMs of the high P´eclet number convection- diﬀusion are illustrated in Fig. 4a, where error is deﬁned as Frobenius distance between HFM and its ROM. For both cases considered, lagrangian ROMs signiﬁcantly outperform the Eulerian ROMs in all cases considered. 4.2. Burger’s equation The proposed approach is next applied to the reduction of a convection-dominated Burger’s equation. Speciﬁcally, we consider (2.1) with f (x, t, w) = w(x, t), f (x, t, w) = − − 1 ∈ × 2 1/Re, w(x, 0) = 0.8 + 0.5 e (x 0.3)2/0.12, w(0, t) = 0, for (x, t) [0, 1.5] [0, 1], where Re = 103. As before, two HFM are constructed, one in the Eulerian frame, as in (2.2), and one in the fully Lagrangian frame, as in (3.2). For both models, a second-order central ﬁnite diﬀerence discretization is used. N = 2000 grid points are used to discretized the (cid:54) (cid:54) domain 0 x 1.5. Total of K = 2000 Eulerian and Lagrangian snapshots are collected. Eulerian and Lagrangian basis are constructed by solving (2.6). Eulerian ROMs are solved in the form of (2.4). Due to the signiﬁcant Lagrangian grid entanglement caused by the nonlinear convection term in the Burger’s equation, the Lagrangian ROMs are solved w w w w ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: R.

text: Mojgani

text: and

text: M.

text: Balajewicz

font: default

text: x

text: x

text: (a)

text: Traditional

text: Eulerian

text: MOR.

text: (b)

text: Lagrangian

text: MOR.

text: Figure

text: Model

text: order

text: reduction

text: of

text: scalar

text: convection

text: equation;

text: HFM

text: (thick

text: grey),

text: k

text: ROMs

text: (dashed

text: green),

text: and

text: k

text: ROMs

text: (solid

text: red).

text: Solutions

text: are

text: plotted

text: for

text: t

text: x

text: x

text: (a)

text: Traditional

text: Eulerian

text: MOR.

text: (b)

text: Lagrangian

text: MOR.

text: Figure

text: Model

text: order

text: reduction

text: of

text: scalar

text: convection-diﬀusion

text: equation

text: with

text: Pe

text: HFM

text: (thick

text: grey),

text: k

text: ROMs

text: (dashed

text: green),

text: and

text: k

text: ROMs

text: (solid

text: red).

text: Solutions

text: are

text: plotted

text: for

text: t

text: traditional

text: Eulerian

text: ROMs

text: and

text: the

text: new,

text: Lagrangian

text: ROMs,

text: are

text: illustrated

text: in

text: in

text: the

text: (a)

text: and

text: (b)

text: subﬁgures.

text: Solutions

text: are

text: plotted

text: for

text: t

text: In

text: both

text: ﬁgures,

text: thick

text: grey

text: lines

text: correspond

text: to

text: the

text: HFM

text: models,

text: while

text: the

text: dashed

text: green

text: and

text: solid

text: red

text: lines

text: correspond

text: to

text: k

text: and

text: k

text: ROMs,

text: respectively.

text: Convergence

text: of

text: Eulerian

text: and

text: Lagrangian

text: ROMs

text: of

text: the

text: high

text: P´eclet

text: number

text: convection-

text: diﬀusion

text: are

text: illustrated

text: in

text: Fig.

text: where

text: error

text: is

text: deﬁned

text: as

text: Frobenius

text: distance

text: between

text: HFM

text: and

text: its

text: ROM.

text: For

text: both

text: cases

text: considered,

text: lagrangian

text: ROMs

text: signiﬁcantly

text: outperform

text: the

text: Eulerian

text: ROMs

text: in

text: all

text: cases

text: considered.

text: Burger’s

text: equation

text: The

text: proposed

text: approach

text: is

text: next

text: applied

text: to

text: the

text: reduction

text: of

text: a

text: convection-dominated

text: Burger’s

text: equation.

text: Speciﬁcally,

text: we

text: consider

text: with

text: f

text: (x,

text: t,

text: w)

text: w(x,

text: t),

text: f

text: (x,

text: t,

text: w)

text: w(x,

text: e

text: (x

text: t)

text: for

text: (x,

text: t)

text: where

text: Re

text: As

text: before,

text: two

text: HFM

text: are

text: constructed,

text: one

text: in

text: the

text: Eulerian

text: frame,

text: as

text: in

text: and

text: one

text: in

text: the

text: fully

text: Lagrangian

text: frame,

text: as

text: in

text: For

text: both

text: models,

text: a

text: second-order

text: central

text: ﬁnite

text: diﬀerence

text: discretization

text: is

text: used.

text: N

text: grid

text: points

text: are

text: used

text: to

text: discretized

text: the

text: domain

text: x

text: Total

text: of

text: K

text: Eulerian

text: and

text: Lagrangian

text: snapshots

text: are

text: collected.

text: Eulerian

text: and

text: Lagrangian

text: basis

text: are

text: constructed

text: by

text: solving

text: Eulerian

text: ROMs

text: are

text: solved

text: in

text: the

text: form

text: of

text: Due

text: to

text: the

text: signiﬁcant

text: Lagrangian

text: grid

text: entanglement

text: caused

text: by

text: the

text: nonlinear

text: convection

text: term

text: in

text: the

text: Burger’s

text: equation,

text: the

text: Lagrangian

text: ROMs

text: are

text: solved

text: w

text: w

text: w

text: w

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##