text: Joint Lensing Analysis of J0850 9 ∆r (kpc) κ 500 1000 2000 4000 −0.2 −0.1 0.0 0.1 0.2 0.3 15'00" 0.25 0.20 10'00" 0.15 05'00" 0.10 0.05 +36°00'00" 0.15 +35°55'00" 0.10 51m00s 40s 20s 50m00s 40s 8h49m20s 0.05 RA (J2000) 0.00 Fig. 9.— Convergence (κ) map from our weak lensing analysis. For visualization purp(cid:48)oses, the map is smooth(cid:48)e×d w(cid:48)ith a circular −0.05 Gaussi∼an of F×WHM 1.5. The ﬁeld of view is 24 24 (correspond- ing to 7.4 7.4 Mpc at the lens redshift) and is centered on the cluster. The color bar indicates the κ at each point in the ﬁeld. −0.10 The massive cluster is detected as the peak near the ﬁeld center. The smaller foreground group is undetected, as its mass is below the scale that can be probed by the weak lensing. −0.15 2 4 8 ∆θ (arcmin) has a greater ellipticity (e.g., ?), with the major axis Fig. 8.— Top: Azimuthally averaged tangential reduced shear roughly pointing toward the multiply-imaged arcs. This (g+) proﬁle as a function of distance from the cluster center. The high ellipticity ((cid:15) = 0.59+−0.11) suggests that the main rising trend of g+ at smaller radius demonstrates the robustness 0.12 of our select◦ion of background galaxies. Bottom: Azimuthally J0850 cluster is an eﬃcient lens (i.e., there should be a averaged 45 -rotated reduced shear (g×) proﬁle as a function of high number density of multiple images) in comparison to distance from the cluster center. The g× component is consistent a spherical halo with an equivalent area enclosed within with a null detection in all bins, as is expected in the absence of systematic errors in the shape measurements. the critical curves (?). The foreground halo at z = 0.2713 has a virial mass × of Mfg = 6.22+−9.21 1013 M(cid:12). The large uncertainties 4. RESULTS arisevbirecause its3.6in2ﬂuence is below the scale probed by 4.1. Constraints from Weak Lensing Analysis the weak lensing, as was seen in Figure ??. However, the weak lensing analysis does place a rough upper limit In Figure ??, we plot the azimuthally averaged pro- ◦ on its mass, as higher mass halos would inﬂuence the ﬁle of the tangential reduced shear (g ) and 45 -rotated + external shear ﬁeld and thus be detected. reduced shear (g×) as a function of distance from the 4.2. Combined Weak and Strong Lensing Constraints cluster center reported by ?. In the weak lensing regime, the shear ﬁeld should be curl-free, so the presence of the We show the constraints from the combined strong and g× component can be used as a check for systematic er- weak lensing analysis in Table ?? (right column). The rors in the shape measurements. Here, the g component distributions are also plotted in Figure ?? for compari- + rises toward the cluster center, as expected, and the g× son with the constraints from weak lensing alone. The component is consistent with zero for all bins. combined analysis produces a main cluster virial mass of × Figure ?? shows the convergence (κ) map derived from M = 2.93+−0.71 1015 M(cid:12), with a tighter uncertainty vir 0.65 our weak lensing analysis using the ? linear inversion than from the weak lensing constraints alone. The in- method (see ?). The massive cluster is clearly detected ferred virial mass is consistent with the dynamical mass near the ﬁeld center. The much smaller foreground halo estimate of ?, conﬁrming that this is among the most is undetected, as its mass is below the scale that can be massive known lensing clusters. The mass of the fore- probed by the weak lensing. ground halo, which has an upper limit from the weak We show the posterior distributions of parameters from lensing analysis, unsurprisingly does not change much our weak lensing analysis in Figure ??. The priors and with the addition of the strong lensing constraint from posterior constraints are also given in Table ?? (center a multiple image pair formed by the main cluster. The column). The virial quantities are deﬁned using the over- galaxy redshift distribution in the J0850 ﬁeld may con- density criterion of ? for our assumed cosmology. tain smaller structures along the line of sight (?), but The main cluster at z = 0.3774 has a virial mass of they are similarly insensitive to the lensing constraints. × M = 3.15+−1.13 1015 M(cid:12), which is in good agreement Compared to the results from weak lensing alone, the vir 0.81 with that determined independently from dynamics by ?. addition of strong lensing information from the single The halo concentration is consistent with that of other multiply-imaged source considerably tightens the con- clusters of comparable mass and redshift (e.g., ?), but straints on the main halo’s concentration by roughly a g g + × )0002J( ceD ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: ASYMPTOTIC FORMULAE FOR MIXED CONGRUENCE STACKS 15 − − − x = e z = e κ+2πi(cid:96)+2πi(a (cid:96)) m m − − = ζk e κ+2πi(a (cid:96)) m m − (cid:48) = ζk e z (3.19) m (cid:48) − − (cid:48) where ζ = e2πi and z = κ 2πi(a (cid:96)). Then we can write F (x) in terms of z : m m m (cid:16) (cid:17) − − F (x) = (xr, xm r; xm)∞1 (3.20) (cid:16) (cid:17) − − (cid:48) − − − (cid:48) − (cid:48) 1 = ζ((cid:96)r (mod m))e rz , ζ( (cid:96)r (mod m))e (m r)z ; e mz m m ∞ − (cid:48) − − (cid:48) (cid:48) 1 (cid:8) =(cid:9) e2πiφ+2πiirz , e 2πiφ+2πii(m r)z ; e2πiimz (3.21) 2π 2π 2π ∞ − (cid:98) (cid:99) where φ := (cid:96)r = (cid:96)r (cid:96)r is the fractional part (this will be nonzero since m m m (cid:16) (cid:17) gcd(r, m) = 1). Then we can express this in terms of theta and eta functions: (cid:0) (cid:1) − − (cid:48) − (cid:48) (cid:48) (cid:48) ieπi imz e πi(φ+irz ) eimz ; eimz 4 2π 2π (cid:0) 2(cid:1)π 2π ∞ F (x) = (cid:48) (cid:48) (cid:0) Θ φ + irz (cid:1), imz 2π 2π − (cid:48) − (cid:48) (cid:48) ieπi imz πi(φ+irz )η imz 6 2π 2π = (cid:48) (cid:48) 2π . (3.22) Θ φ + irz , imz 2π 2π Now using the transformations of the theta (2.7) and eta (2.8) functions in (3.22), we have (cid:0) (cid:1) (cid:0) (cid:1) −eπi(iπz(cid:48) (1m2−r2)−φ)+m2πz2(cid:48) (φ2+φirz(cid:48)−r42πz2(cid:48)2 )η −2π(cid:48) F (x) = − imz . (3.23) Θ 2πi(cid:48)φ + r ; 2πi(cid:48) mz m mz (cid:18) (cid:19) For the eta function, the approximation is − η 2π(cid:48) ∼ e1π2i·i−m2zπ(cid:48) = e6−mπz2(cid:48) (3.24) imz ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: ASYMPTOTIC

label: paragraph

text: FORMULAE

text: FOR

text: MIXED

text: CONGRUENCE

text: STACKS

label: equation

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text: e

text: z

text: e

text: m

text: m

text: ζk

text: e

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text: m

text: ζk

text: e

text: z

text: m

text: where

text: ζ

text: and

text: z

text: κ

text: Then

text: we

text: can

text: write

text: F

text: (x)

text: in

text: terms

text: of

text: z

text: m

text: m

text: m

text: F

text: (x)

text: (xr,

text: xm

text: r;

text: (mod

text: m))e

text: rz

text: ζ(

text: (mod

text: m))e

text: (m

text: r)z

text: e

text: mz

text: m

text: m

text: e

text: r)z

text: 2π

text: 2π

text: 2π

text: where

text: φ

text: is

text: the

text: fractional

text: part

text: (this

text: will

text: be

text: nonzero

text: since

text: m

text: m

text: m

text: gcd(r,

text: m)

text: Then

text: we

text: can

text: express

text: this

text: in

text: terms

text: of

text: theta

text: and

text: eta

text: functions:

text: ieπi

text: imz

text: e

text: πi(φ+irz

text: eimz

text: eimz

text: 2π

text: 2π

text: 2π

text: F

text: (x)

text: Θ

text: φ

text: irz

text: imz

text: 2π

text: 2π

text: ieπi

text: imz

text: πi(φ+irz

text: )η

text: imz

text: 2π

text: 2π

text: 2π

text: Θ

text: φ

text: irz

text: imz

text: 2π

text: 2π

text: Now

text: using

text: the

text: transformations

text: of

text: the

text: theta

text: and

text: eta

text: functions

text: in

text: we

text: have

text: )η

text: F

text: (x)

text: imz

text: Θ

text: r

text: mz

text: m

text: mz

text: For

text: the

text: eta

text: function,

text: the

text: approximation

text: is

text: η

text: imz

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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 RICHARD FRNKA (cid:88) (cid:88) (cid:0) (cid:1) qn − 1 + − = (1 + a) a3(cid:88)nq n2 (3n+1) 1 a2q2n+1 (aq; q) (a 1q; q) ≥ n n ≥ n 1 n 0 − − ( 1)na2nq n(n+1) 2 a − . (2.2) (aq; q)∞(a 1q; q)∞ ≥ n 0 This equation is a special case of Heine’s transformation formula for hypergeometric (cid:55)→ (cid:55)→ − − series. Sending q qm and a q r, multiplying by −qr and reorganizing in (2.2) (1 qr) (cid:3) completes the proof. We wish to analyze individually the diﬀerent pieces of the right hand side of (2.1), so we let S (q) = F (q)L(q)(cid:88)+ R(q) (r,m) where F (q) :=(qr, qm−r(cid:88); qm)−∞1, L(q) := (−(cid:0)1)nq mn(2n+1)−2rn,(cid:1) ≥ n 0 − − − − − and R(q) := ( 1)n 1q mn(3n+1) 3rn 1 q(2n+1)m 2r 2 ≥ (cid:80) n 0 ± The R(q) term only oﬀers a contribution of 1 for high order terms, so letting H(q) := ∼ F (q)L(q) be the main term, we have that s (n) h(n) for H(q) = ≥ h(n)qn. (r,m) n 1 With the generating series in this particular form, we can turn our attention to each individual piece. The sum L(q) is a false theta function whose coeﬃcients can be bounded, and the inﬁnite q-series satisﬁes a modular transformation formula. The transformation formula will provide a ﬁrst term in the asymptotic expansion of F (q) that will be vital for the circle and function over which we will be integrating. → Proposition 2.2. For q = e2πiτ and 0 < r < m/2, as τ (cid:16) 0, (cid:16) (cid:17)(cid:17) 1 1 − − − − = − e( πirm(m r)+πi6m)τ+6πmiτ 1 + O e m2πτi . (2.3) (qr, qm r; qm)∞ 2 sin(π(m r)) m Proof. The function F (q) can be expressed in terms of Jacobi’s theta function (see [10] for a good reference), which is ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RICHARD

text: FRNKA

text: qn

text: a)

text: (aq;

text: q)

text: (a

text: q)

text: n

text: n

text: n

text: n

text: a

text: (aq;

text: q)∞(a

text: q)∞

text: n

text: This

text: equation

text: is

text: a

text: special

text: case

text: of

text: Heine’s

text: transformation

text: formula

text: for

text: hypergeometric

text: series.

text: Sending

text: q

text: qm

text: and

text: a

text: q

text: r,

text: multiplying

text: by

text: −qr

text: and

text: reorganizing

text: in

text: qr)

text: completes

text: the

text: proof.

text: We

text: wish

text: to

text: analyze

text: individually

text: the

text: diﬀerent

text: pieces

text: of

text: the

text: right

text: hand

text: side

text: of

text: so

text: we

text: let

text: S

text: (q)

text: F

text: R(q)

text: (r,m)

text: where

text: F

text: (q)

text: :=(qr,

text: L(q)

text: n

text: and

text: R(q)

text: n

text: The

text: R(q)

text: term

text: only

text: oﬀers

text: a

text: contribution

text: of

text: for

text: high

text: order

text: terms,

text: so

text: letting

text: H(q)

text: F

text: (q)L(q)

text: be

text: the

text: main

text: term,

text: we

text: have

text: that

text: s

text: (n)

text: h(n)

text: for

text: H(q)

text: h(n)qn.

text: (r,m)

text: n

text: With

text: the

text: generating

text: series

text: in

text: this

text: particular

text: form,

text: we

text: can

text: turn

text: our

text: attention

text: to

text: each

text: individual

text: piece.

text: The

text: sum

text: L(q)

text: is

text: a

text: false

text: theta

text: function

text: whose

text: coeﬃcients

text: can

text: be

text: bounded,

text: and

text: the

text: inﬁnite

text: q-series

text: satisﬁes

text: a

text: modular

text: transformation

text: formula.

text: The

text: transformation

text: formula

text: will

text: provide

text: a

text: ﬁrst

text: term

text: in

text: the

text: asymptotic

text: expansion

text: of

text: F

text: (q)

text: that

text: will

text: be

text: vital

text: for

text: the

text: circle

text: and

text: function

text: over

text: which

text: we

text: will

text: be

text: integrating.

text: Proposition

text: For

text: q

text: and

text: r

text: as

text: τ

text: e(

text: πirm(m

text: O

text: e

text: (qr,

text: qm

text: r;

text: qm)∞

text: sin(π(m

text: r))

text: m

text: Proof.

text: The

text: function

text: F

text: (q)

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: Jacobi’s

text: theta

text: function

text: (see

text: for

text: a

text: good

text: reference),

text: which

text: is

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: ASYMPTOTIC FORMULAE FOR MIXED CONGRUENCE STACKS 9 | | − | | → Lemma 2.4. For arg(z) < π c where c is some arbitrary constant, taking z 0 2 gives (cid:88) − − k 1 L(x) = α α zs + O(zk), (2.17) 0 s s=1 where − ∼ (4r + m)(ms 1) ≤ ≤ − 1 α for 1 s k 1, and α := . s 22s+1 0 2 3. Wright’s Circle Method and Proof of Theorem 1.1 − The proof follows the notation and method of Wright in [21]. We let x = e z (cid:55)→ replace q = e2πiτ and send τ iz . Then the modular transformation in Proposition 2π (cid:16) (cid:16) (cid:17)(cid:17) 2.2 becomes (cid:0) (cid:1) (cid:16) (cid:17) 1 − m + r2 −r z+ π2 −4π2 F (x) = e 12 2m 2 3mz 1 + O e mz . (3.1) 2 sin πr m (cid:0) (cid:1) Deﬁning w(z) := 1 e(r(m2m−r)−1m2)z+3πm2z , 2 sin πr m (cid:0) (cid:1) the following term is critical for the Circle Method w(z)x−n = 1 e(r(m2m−r)−1m2+n)z+3πm2z . 2 sin πr (cid:115) (cid:18) (cid:19) m (cid:18) (cid:19) Using the Arithmetic-Geometric Mean Inequality on the exponent above gives − − r(m r) − m π2 ≥ π2 r(m r) − m + n z + 2 + n , 2m 12 3mz 3m 2m 12 (cid:113) where equality occurs when z = κ with π κ := . − − 3r(m r) m2 + 3mn 2 4 − The circle C of radius e κ centered at 0 will be the contour of integration. The | | ≤ major arc C is deﬁned to be where arg(x) ρκ for small ﬁxed ρ > 0. The minor 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: ASYMPTOTIC

text: FORMULAE

text: FOR

text: MIXED

text: CONGRUENCE

text: STACKS

text: Lemma

text: For

text: arg(z)

text: π

text: c

text: where

text: c

text: is

text: some

text: arbitrary

text: constant,

text: taking

text: z

text: gives

text: k

text: L(x)

text: α

text: α

text: zs

text: O(zk),

text: s

text: where

text: m)(ms

text: α

text: for

text: s

text: k

text: and

text: α

text: s

label: section

text: Wright’s

text: Circle

text: Method

text: and

text: Proof

text: of

text: Theorem

text: The

text: proof

text: follows

text: the

text: notation

text: and

text: method

text: of

text: Wright

text: in

text: We

text: let

text: x

text: e

text: z

text: replace

text: q

text: and

text: send

text: τ

text: iz

text: Then

text: the

text: modular

text: transformation

text: in

text: Proposition

text: 2π

text: becomes

text: m

text: −r

text: z+

text: π2

text: −4π2

text: F

text: (x)

text: e

text: O

text: e

text: mz

text: sin

text: πr

text: m

text: Deﬁning

text: w(z)

text: sin

text: πr

text: m

text: the

text: following

text: term

text: is

text: critical

text: for

text: the

text: Circle

text: Method

text: w(z)x−n

text: sin

text: πr

text: m

text: Using

text: the

text: Arithmetic-Geometric

text: Mean

text: Inequality

text: on

text: the

text: exponent

text: above

text: gives

text: r(m

text: r)

text: m

text: π2

text: π2

text: r(m

text: r)

text: m

text: n

text: z

text: n

text: where

text: equality

text: occurs

text: when

text: z

text: κ

text: with

text: π

text: κ

text: r)

text: The

text: circle

text: C

text: of

text: radius

text: e

text: κ

text: centered

text: at

text: will

text: be

text: the

text: contour

text: of

text: integration.

text: The

text: major

text: arc

text: C

text: is

text: deﬁned

text: to

text: be

text: where

text: arg(x)

text: ρκ

text: for

text: small

text: ﬁxed

text: ρ

text: The

text: minor

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: ##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: | | | | the case of ALPs, B = 5(T). We use, for simplicity, G = 1 for mW < ω and G = 0 for 0 0 mW > ω . 0 The results of this proposal seem very promising, because the projected sensitivity of such setup covers a wide space of parameters not yet explored. However, this calculation does not consider details in a realistic experimental setup. For instance, it is important that the frequency of the oscillating wall is stable and ﬂuctuations in frequency must be small enough, so that we do not depart from the resonance condition (13). In reference [8], in the context of the Dynamical Casimir Eﬀect, the authors propose an experimental technique, where they achieve to make an eﬀective motion of the wall in the order of GHz and ﬂuctuations less than 100Hz. The latter could be a possibility to apply the idea proposed in this paper. Acknowledgement We would like to thank Paola Arias for her help and useful comments. This work was supported by USA-1555 (A. A.). References [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977); S. Weinberg, Phys. Rev. Lett. 40, 223 (1978); F. Wilczek, Phys. Rev. Lett. 40, 279 (1978). [2] P. Svrcek and E. Witten, JHEP 0606, 051 (2006); E. Witten, Phys. Lett. B 149, 351 (1984);. J. P. Conlon, JHEP 0605, 078 (2006); A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, Phys. Rev. D 81, 123530 (2010). [3] B. Holdom, Phys. Lett. B 166, 196 (1986); C. Boehm and P. Fayet, Nucl. Phys. B 683, 219 (2004); P. Fayet, Phys. Rev. D 75, 115017 (2007); M. Pospelov, A. Ritz and M. B. Voloshin, Phys. Lett. B 662, 53 (2008). [4] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983); P. Sikivie, Phys. Rev. D 32, 2988 (1985); L. Krauss, J. Moody, F. Wilczek and D. E. Morris, Phys. Rev. Lett. 55, 1797 (1985); S. J. Asztalos et al. [ADMX Collaboration], Phys. Rev. Lett. 104, 041301 (2010); S. Andriamonje et al. [CAST Collaboration], JCAP 0704, 010 (2007); E. Arik et al. [CAST Collaboration], JCAP 0902, 008 (2009); Y. Inoue, Y. Akimoto, R. Ohta, T. Mizumoto, A. Yamamoto and M. Minowa, Phys. Lett. B 668, 93 (2008); A. Wagner et al. [ADMX Collaboration], Phys. Rev. Lett. 105, 171801 (2010). [5] K. Van Bibber, N. R. Dagdeviren, S. E. Koonin, A. Kerman and H. N. Nelson, Phys. Rev. Lett. 59, 759 (1987). [6] F. Hoogeveen and T. Ziegenhagen, Nucl. Phys. B 358, 3 (1991); R. Cameron et al., Phys. Rev. D 47, 3707 (1993); K. Ehret et al., Phys. Lett. B 689, 149 (2010); P. Sikivie, D. B. Tanner and K. van Bibber, Phys. Rev. Lett. 98, 172002 (2007); P. Arias, J. Jaeckel, J. Redondo and A. Ringwald, Phys. Rev. D 82, 115018 (2010). 7

text: the

text: case

text: of

text: ALPs,

text: B

text: We

text: use,

text: for

text: simplicity,

text: G

text: for

text: mW

text: ω

text: and

text: G

text: for

text: mW

text: ω

text: The

text: results

text: of

text: this

text: proposal

text: seem

text: very

text: promising,

text: because

text: the

text: projected

text: sensitivity

text: of

text: such

text: setup

text: covers

text: a

text: wide

text: space

text: of

text: parameters

text: not

text: yet

text: explored.

text: However,

text: this

text: calculation

text: does

text: not

text: consider

text: details

text: in

text: a

text: realistic

text: experimental

text: setup.

text: For

text: instance,

text: it

text: is

text: important

text: that

text: the

text: frequency

text: of

text: the

text: oscillating

text: wall

text: is

text: stable

text: and

text: ﬂuctuations

text: in

text: frequency

text: must

text: be

text: small

text: enough,

text: so

text: that

text: we

text: do

text: not

text: depart

text: from

text: the

text: resonance

text: condition

text: In

text: reference

text: in

text: the

text: context

text: of

text: the

text: Dynamical

text: Casimir

text: Eﬀect,

text: the

text: authors

text: propose

text: an

text: experimental

text: technique,

text: where

text: they

text: achieve

text: to

text: make

text: an

text: eﬀective

text: motion

text: of

text: the

text: wall

text: in

text: the

text: order

text: of

text: GHz

text: and

text: ﬂuctuations

text: less

text: than

text: The

text: latter

text: could

text: be

text: a

text: possibility

text: to

text: apply

text: the

text: idea

text: proposed

text: in

text: this

text: paper.

text: Acknowledgement

text: We

text: would

text: like

text: to

text: thank

text: Paola

text: Arias

text: for

text: her

text: help

text: and

text: useful

text: comments.

text: This

text: work

text: was

text: supported

text: by

text: (A.

text: A.).

text: References

text: R.

text: D.

text: Peccei

text: and

text: H.

text: R.

text: Quinn,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Weinberg,

text: Phys.

text: Rev.

text: Lett.

text: F.

text: Wilczek,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Svrcek

text: and

text: E.

text: Witten,

text: JHEP

text: E.

text: Witten,

text: Phys.

text: Lett.

text: B

text: J.

text: P.

text: Conlon,

text: JHEP

text: A.

text: Arvanitaki,

text: S.

text: Dimopoulos,

text: S.

text: Dubovsky,

text: N.

text: Kaloper

text: and

text: J.

text: March-Russell,

text: Phys.

text: Rev.

text: D

text: B.

text: Holdom,

text: Phys.

text: Lett.

text: B

text: C.

text: Boehm

text: and

text: P.

text: Fayet,

text: Nucl.

text: Phys.

text: B

text: P.

text: Fayet,

text: Phys.

text: Rev.

text: D

text: M.

text: Pospelov,

text: A.

text: Ritz

text: and

text: M.

text: B.

text: Voloshin,

text: Phys.

text: Lett.

text: B

text: P.

text: Sikivie,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Sikivie,

text: Phys.

text: Rev.

text: D

text: L.

text: Krauss,

text: J.

text: Moody,

text: F.

text: Wilczek

text: and

text: D.

text: E.

text: Morris,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: J.

text: Asztalos

text: et

text: al.

text: [ADMX

text: Collaboration],

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Andriamonje

text: et

text: al.

text: [CAST

text: Collaboration],

text: JCAP

text: E.

text: Arik

text: et

text: al.

text: [CAST

text: Collaboration],

text: JCAP

text: Y.

text: Inoue,

text: Y.

text: Akimoto,

text: R.

text: Ohta,

text: T.

text: Mizumoto,

text: A.

text: Yamamoto

text: and

text: M.

text: Minowa,

text: Phys.

text: Lett.

text: B

text: A.

text: Wagner

text: et

text: al.

text: [ADMX

text: Collaboration],

text: Phys.

text: Rev.

text: Lett.

text: K.

text: Van

text: Bibber,

text: N.

text: R.

text: Dagdeviren,

text: S.

text: E.

text: Koonin,

text: A.

text: Kerman

text: and

text: H.

text: N.

text: Nelson,

text: Phys.

text: Rev.

text: Lett.

text: F.

text: Hoogeveen

text: and

text: T.

text: Ziegenhagen,

text: Nucl.

text: Phys.

text: B

text: R.

text: Cameron

text: et

text: al.,

text: Phys.

text: Rev.

text: D

text: K.

text: Ehret

text: et

text: al.,

text: Phys.

text: Lett.

text: B

text: P.

text: Sikivie,

text: D.

text: B.

text: Tanner

text: and

text: K.

text: van

text: Bibber,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Arias,

text: J.

text: Jaeckel,

text: J.

text: Redondo

text: and

text: A.

text: Ringwald,

text: Phys.

text: Rev.

text: D


text: 7 − − − − − − − − − − 1( Aae at Bme mt cos(nt) Bne mt sin(nt) (Cm/n)e mt sin(nt) Ce mt cos(nt)) Us(k, τ, t) = 2 − − − (A32) Ae at + Be mt cos(nt) + (C/n)e mt sin(nt) + D In the absence of carrier scattering, and zero static electric and magnetic ﬁelds, the Boltzmann equation simpliﬁes to ∂f − s,k = Us(k, τ, t)(f (cid:48) f ) (A33) ∂t s ,k s,k For the conduction band, the above equation is ∂f − c,k = Uc(k, τ, t)(1 2f ) = W(k, τ, t) (A34) ∂t c,k as expected from the solution to the Bloch equations. Using a phenomenological description for carrier relaxation we have − I −f f0 (f ) = s,k s,k (A35) s,k τ s,k where f0 is the equilibrium distribution function, and τ s,k s,k is the phenomenological intra-band scattering time. Note that now the equilibrium distribution function f0 must be band → ∞ → s,k dependent. When t , U(k, τ, t) 0, as a steady state has been reached, with hole and electron type carriers in the valence band and conduction band respectively. The function f0 can be identiﬁed with the electron distribution in the s s,k band for a ﬁnite carrier distribution created by the laser i.e. ≡ → ∞ ≡ → ∞ f0 ρ (t ) and f0 ρ (t ), which holds c,k cc →v,k ∞ vv exactly true in the limit τ . For a generic τ value this might not hold true always. Therefore, for numerical calcu- lation of f , we start with an initial guess of f0, and then k → ∞ compare f (t ) with the guess f0, and re-evaluate f k → ∞ → k until the condition f (t ) f0 is satisﬁed. k The following equation describes the distribution function in the conduction band FIG. 5. The electron distribution function f at a valley point as a K − function of time obtained from the Boltzmann equation, for various ∂f∂ct,k + k˙ · ∂∂fck,k = −fc,kτc,kfc0,k + U(k, τ, t)(1 − 2fc,k) vteadluienstohfesacbatsteenricnegotfimanesy(sτc)a.ttIenrirnegd i(sτth→e d∞istr)i.buFtoiornlafrugnectτiovnapluloest-, (A36) the electron distribution function approaches the red curve. We use 1/Γ = 5ps, ω = 1meV , δ = 0.1ω . RK RK We ﬁrst discuss solutions of the above equation in the ab- sence of any static electric ﬁelds. i.e. k˙ = 0. We now have − two rates (τ 1 and U(k, τ, t)) which govern the dynamics of c,k fc, given by Appendix B: The solution to Semiconductor Bloch Equations k (SBEs) in TMD − ∂fc,k = −fc,k fc0,k + U(k, τ, t)(1 − 2f ) (A37) Here we will discuss the general solution to SBEs relevant ∂t τc,k c,k to TMDs. The total Hamiltonian (Hk + HI) described in the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Aae

text: at

text: Bme

text: mt

text: cos(nt)

text: Bne

text: mt

text: sin(nt)

text: (Cm/n)e

text: mt

text: sin(nt)

text: Ce

text: mt

text: cos(nt))

text: Us(k,

text: τ,

text: t)

text: Ae

text: at

text: Be

text: mt

text: cos(nt)

text: (C/n)e

text: mt

text: sin(nt)

text: D

text: In

text: the

text: absence

text: of

text: carrier

text: scattering,

text: and

text: zero

text: static

text: electric

text: and

text: magnetic

text: ﬁelds,

text: the

text: Boltzmann

text: equation

text: simpliﬁes

text: to

text: ∂f

text: s,k

text: Us(k,

text: τ,

text: t)(f

text: f

text: ∂t

text: s

text: ,k

text: s,k

text: For

text: the

text: conduction

text: band,

text: the

text: above

text: equation

text: is

text: ∂f

text: c,k

text: Uc(k,

text: τ,

text: W(k,

text: τ,

text: t)

text: ∂t

text: c,k

text: as

text: expected

text: from

text: the

text: solution

text: to

text: the

text: Bloch

text: equations.

text: Using

text: a

text: phenomenological

text: description

text: for

text: carrier

text: relaxation

text: we

text: have

text: I

text: −f

text: (f

text: s,k

text: s,k

text: s,k

text: τ

text: s,k

text: where

text: is

text: the

text: equilibrium

text: distribution

text: function,

text: and

text: τ

text: s,k

text: s,k

text: is

text: the

text: phenomenological

text: intra-band

text: scattering

text: time.

text: Note

text: that

text: now

text: the

text: equilibrium

text: distribution

text: function

text: must

text: be

text: band

text: s,k

text: dependent.

text: When

text: t

text: U(k,

text: τ,

text: t)

text: as

text: a

text: steady

text: state

text: has

text: been

text: reached,

text: with

text: hole

text: and

text: electron

text: type

text: carriers

text: in

text: the

text: valence

text: band

text: and

text: conduction

text: band

text: respectively.

text: The

text: function

text: can

text: be

text: identiﬁed

text: with

text: the

text: electron

text: distribution

text: in

text: the

text: s

text: s,k

text: band

text: for

text: a

text: ﬁnite

text: carrier

text: distribution

text: created

text: by

text: the

text: laser

text: i.e.

text: ρ

text: (t

text: and

text: ρ

text: (t

text: which

text: holds

text: c,k

text: cc

text: →v,k

text: vv

text: exactly

text: true

text: in

text: the

text: limit

text: τ

text: For

text: a

text: generic

text: τ

text: value

text: this

text: might

text: not

text: hold

text: true

text: always.

text: Therefore,

text: for

text: numerical

text: calcu-

text: lation

text: of

text: f

text: we

text: start

text: with

text: an

text: initial

text: guess

text: of

text: and

text: then

text: k

text: compare

text: f

text: (t

text: with

text: the

text: guess

text: and

text: re-evaluate

text: f

text: k

text: k

text: until

text: the

text: condition

text: f

text: (t

text: is

text: satisﬁed.

text: k

text: The

text: following

text: equation

text: describes

text: the

text: distribution

text: function

text: in

text: the

text: conduction

text: band

text: FIG.

text: The

text: electron

text: distribution

text: function

text: f

text: at

text: a

text: valley

text: point

text: as

text: a

text: K

text: function

text: of

text: time

text: obtained

text: from

text: the

text: Boltzmann

text: equation,

text: for

text: various

text: ∂f∂ct,k

text: k˙

text: ∂∂fck,k

text: U(k,

text: τ,

text: i(sτth→e

text: the

text: electron

text: distribution

text: function

text: approaches

text: the

text: red

text: curve.

text: We

text: use

text: ω

text: δ

text: RK

text: RK

text: We

text: ﬁrst

text: discuss

text: solutions

text: of

text: the

text: above

text: equation

text: in

text: the

text: ab-

text: sence

text: of

text: any

text: static

text: electric

text: ﬁelds.

text: i.e.

text: k˙

text: We

text: now

text: have

text: two

text: rates

text: (τ

text: and

text: U(k,

text: τ,

text: t))

text: which

text: govern

text: the

text: dynamics

text: of

text: c,k

text: fc,

text: given

text: by

text: Appendix

text: B:

text: The

text: solution

text: to

text: Semiconductor

text: Bloch

text: Equations

text: k

text: (SBEs)

text: in

text: TMD

text: ∂fc,k

text: −fc,k

text: U(k,

text: τ,

text: Here

text: we

text: will

text: discuss

text: the

text: general

text: solution

text: to

text: SBEs

text: relevant

text: ∂t

text: τc,k

text: c,k

text: to

text: TMDs.

text: The

text: total

text: Hamiltonian

text: (Hk

text: HI)

text: described

text: in

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: A short-orbit spectrometer for low-energy pion detection in electroproduction experiments at MAMI D. Baumann1, M. Ding1, I. Frisˇcˇic´1, R. Bo¨hm1, D. Bosnar1, M. O. Distler1, H. Merkel1, U. Mu¨ller1, Th. Walcher1, M. Wendel1 aInstitut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t, 55099 Mainz, Germany bDepartment of Physics, Faculty of Science, University of Zagreb, Bijenicˇka c. 32, 10000 Zagreb, Croatia Abstract A new Short-Orbit Spectrometer (SOS) has been constructed and installed within the experimental facility of the A1 collaboration at Mainz Microtron (MAMI), with the goal to detect low-energy pions. It is equipped with a Browne-Buechner magnet and a detector system consisting of two helium-ethane based drift chambers and a scintillator telescope made of ﬁve layers. The detector system allows detection of pions in the momentum range of 50 – 147 MeV/c, which corresponds to 8.7 – 63 MeV kinetic energy. The spectrometer can be placed at a distance range of 54 – 66 cm from the target center. Two collimators are available for the mea- surements, one having 1.8 msr aperture and the other having 7 msr aperture. The Short-Orbit Spectrometer has been successfully calibrated and used in coincidence measurements together with the standard magnetic spectrometers of the A1 collaboration. Keywords: Short-orbit spectrometer, Low-energy pions, Electroproduction experiments PACS: 29.30.-h (Spectrometers for nuclear physics), 25.30.Fj (Inelastic electron scattering in nuclear reactions) 1. Introduction particles having momenta up to 735 MeV/c in A, 870 MeV/c in B and 551 MeV/c in C. For unstable particles the use of spec- (cid:48) The p(e, e π+)n reaction, measured with high precision close trometers is limited to high momenta, which allow the particles to threshold, provides a great tool for studying the axial and to reach the detector system before they decay. the pseudoscalar form factor [? ? ]. One of the experimental The pion is an unstable particle having a lifetime of 26.033 setups capable performing such experiments is the high resolu- ns, which dominantly decays into a muon and a muon neutrino tion spectrometer setup of the A1 collaboration [? ] using the [? ]. Close to the reaction threshold it will have low kinetic electron accelerator Mainz Microtron (MAMI) [? ]. energy. Since the distance from target to standard detector sys- The A1 spectrometer setup consists of two large accep- tem inside the spectrometers of the A1 collaboration is of the tance (28 msr) spectrometers (A and C) having a quadrupole- order of 10 m, most of the low energy pions will decay be- sextupole-dipole-dipole magnet conﬁguration and a third spec- fore they reach the detectors. The measurement time needed trometer (B), which is based on one clamshell dipole. This for collection of statistically relevant data would be prolonged. spectrometer is slimmer and can reach scattering angles down Moreover, a fraction of muons will be created close to the di- ◦ to 7 , but this comes at the cost of smaller acceptance (5.6 msr). rection of pions and will be also detected. Such muons cannot Each spectrometer is equipped with a particle tracking system be distinguished from the pions and they are a major source based on 4 vertical drift chamber layers and a particle iden- of systematic error. The inﬂuence of these two problems can tiﬁcation system based on scintillating paddles and Cˇ erenkov be reduced simultaneously by decreasing the distance between detectors. If needed, these can be replaced with other detec- target and detector system. Signiﬁcantly more pions will now tor conﬁgurations. The spectrometers can be rotated around the survive to the detector system for the same kinetic energy, but target chamber in the center and they can be operated in a sin- the data still has to be corrected for pion decay. At the same gle, double, and triple coincidence mode. Looking along the time, muon contamination of the data will be reduced, so that it beam direction, spectrometers A and C are placed on the left is feasible to determine it from a simulation and correct for it. and right side of the beam pipe, respectively. Spectrometer B Therefore, a new short orbit spectrometer (SOS) was con- can be moved to either side of the beam pipe; in standard con- structed speciﬁcally for detection of low energy pions. The (cid:48) ﬁhagvueraatiroenlaittivies mploacmeednotunmthreesroigluhttiosind≤e.1A0l−l4thanredeasrpeeucsterodminedteirfs- 1th2Ce (ﬁer,set pph)1y1sBicrseeaxcptieornimweanst,utsheedpf(oer, cea(cid:48)πli+b)rnatrieoanctoifonthwe aSsOeSv.aAlus- ferent electron scattering experiments for detection of charged ated, which at the same time allowed to test the performance of the SOS in detection of low energy pions. Section ?? of this paper describes the magnet, detector sys- ∗ Corresponding author, Email: friscic@mit.edu tem and electronics of the SOS. In section ?? the measurement 1Present address: MIT-LNS, Cambridge MA, 02139, USA used for calibration of the SOS is discussed. Section ?? deals 2This paper comprises parts of the doctoral theses of D. Baumann, M. Ding (cid:48) and I. Frisˇcˇic´. with a measurement of the p(e, e π+)n reaction. A summary and Preprint submitted to Nucl. Instr. and Meth. A May 30, 2020 ##LTLine## ##LTLine## ##LTLine##

text: A

label: title

text: short-orbit

text: spectrometer

text: for

text: low-energy

text: pion

text: detection

text: in

text: electroproduction

text: experiments

text: at

text: MAMI

text: D.

label: author

text: M.

text: I.

text: R.

text: D.

text: M.

text: O.

text: H.

text: U.

text: Th.

text: M.

text: aInstitut

text: fu¨r

text: Kernphysik,

text: Johannes

text: Gutenberg-Universita¨t,

text: Mainz,

text: Germany

text: bDepartment

text: of

text: Physics,

text: Faculty

text: of

text: Science,

text: University

text: of

text: Zagreb,

text: Bijenicˇka

text: c.

text: Zagreb,

text: Croatia

text: Abstract

text: A

label: abstract

text: new

text: Short-Orbit

text: Spectrometer

text: (SOS)

text: has

text: been

text: constructed

text: and

text: installed

text: within

text: the

text: experimental

text: facility

text: of

text: the

text: collaboration

text: at

text: Mainz

text: Microtron

text: (MAMI),

text: with

text: the

text: goal

text: to

text: detect

text: low-energy

text: pions.

text: It

text: is

text: equipped

text: with

text: a

text: Browne-Buechner

text: magnet

text: and

text: a

text: detector

text: system

text: consisting

text: of

text: two

text: helium-ethane

text: based

text: drift

text: chambers

text: and

text: a

text: scintillator

text: telescope

text: made

text: of

text: ﬁve

text: layers.

text: The

text: detector

text: system

text: allows

text: detection

text: of

text: pions

text: in

text: the

text: momentum

text: range

text: of

text: MeV/c,

text: which

text: corresponds

text: to

text: MeV

text: kinetic

text: energy.

text: The

text: spectrometer

text: can

text: be

text: placed

text: at

text: a

text: distance

text: range

text: of

text: cm

text: from

text: the

text: target

text: center.

text: Two

text: collimators

text: are

text: available

text: for

text: the

text: mea-

text: surements,

text: one

text: having

text: msr

text: aperture

text: and

text: the

text: other

text: having

text: msr

text: aperture.

text: The

text: Short-Orbit

text: Spectrometer

text: has

text: been

text: successfully

text: calibrated

text: and

text: used

text: in

text: coincidence

text: measurements

text: together

text: with

text: the

text: standard

text: magnetic

text: spectrometers

text: of

text: the

text: collaboration.

text: Keywords:

text: Short-orbit

text: spectrometer,

text: Low-energy

text: pions,

text: Electroproduction

text: experiments

text: PACS:

text: (Spectrometers

text: for

text: nuclear

text: physics),

text: (Inelastic

text: electron

text: scattering

text: in

text: nuclear

text: reactions)

text: Introduction

text: particles

text: having

text: momenta

text: up

text: to

text: MeV/c

text: in

text: A,

text: MeV/c

text: in

text: B

text: and

text: MeV/c

text: in

text: C.

text: For

text: unstable

text: particles

text: the

text: use

text: of

text: spec-

text: The

text: p(e,

text: e

text: π+)n

text: reaction,

text: measured

text: with

text: high

text: precision

text: close

text: trometers

text: is

text: limited

text: to

text: high

text: momenta,

text: which

text: allow

text: the

text: particles

text: to

text: threshold,

text: provides

text: a

text: great

text: tool

text: for

text: studying

text: the

text: axial

text: and

text: to

text: reach

text: the

text: detector

text: system

text: before

text: they

text: decay.

text: the

text: pseudoscalar

text: form

text: factor

text: One

text: of

text: the

text: experimental

text: The

text: pion

text: is

text: an

text: unstable

text: particle

text: having

text: a

text: lifetime

text: of

text: setups

text: capable

text: performing

text: such

text: experiments

text: is

text: the

text: high

text: resolu-

text: ns,

text: which

text: dominantly

text: decays

text: into

text: a

text: muon

text: and

text: a

text: muon

text: neutrino

text: tion

text: spectrometer

text: setup

text: of

text: the

text: collaboration

text: using

text: the

text: Close

text: to

text: the

text: reaction

text: threshold

text: it

text: will

text: have

text: low

text: kinetic

text: electron

text: accelerator

text: Mainz

text: Microtron

text: (MAMI)

text: energy.

text: Since

text: the

text: distance

text: from

text: target

text: to

text: standard

text: detector

text: sys-

text: The

text: spectrometer

text: setup

text: consists

text: of

text: two

text: large

text: accep-

text: tem

text: inside

text: the

text: spectrometers

text: of

text: the

text: collaboration

text: is

text: of

text: the

text: tance

text: msr)

text: spectrometers

text: (A

text: and

text: C)

text: having

text: a

text: quadrupole-

text: order

text: of

text: m,

text: most

text: of

text: the

text: low

text: energy

text: pions

text: will

text: decay

text: be-

text: sextupole-dipole-dipole

text: magnet

text: conﬁguration

text: and

text: a

text: third

text: spec-

text: fore

text: they

text: reach

text: the

text: detectors.

text: The

text: measurement

text: time

text: needed

text: trometer

text: (B),

text: which

text: is

text: based

text: on

text: one

text: clamshell

text: dipole.

text: This

text: for

text: collection

text: of

text: statistically

text: relevant

text: data

text: would

text: be

text: prolonged.

text: spectrometer

text: is

text: slimmer

text: and

text: can

text: reach

text: scattering

text: angles

text: down

text: Moreover,

text: a

text: fraction

text: of

text: muons

text: will

text: be

text: created

text: close

text: to

text: the

text: di-

text: to

text: but

text: this

text: comes

text: at

text: the

text: cost

text: of

text: smaller

text: acceptance

text: msr).

text: rection

text: of

text: pions

text: and

text: will

text: be

text: also

text: detected.

text: Such

text: muons

text: cannot

text: Each

text: spectrometer

text: is

text: equipped

text: with

text: a

text: particle

text: tracking

text: system

text: be

text: distinguished

text: from

text: the

text: pions

text: and

text: they

text: are

text: a

text: major

text: source

text: based

text: on

text: vertical

text: drift

text: chamber

text: layers

text: and

text: a

text: particle

text: iden-

text: of

text: systematic

text: error.

text: The

text: inﬂuence

text: of

text: these

text: two

text: problems

text: can

text: tiﬁcation

text: system

text: based

text: on

text: scintillating

text: paddles

text: and

text: Cˇ

text: erenkov

text: be

text: reduced

text: simultaneously

text: by

text: decreasing

text: the

text: distance

text: between

text: detectors.

text: If

text: needed,

text: these

text: can

text: be

text: replaced

text: with

text: other

text: detec-

text: target

text: and

text: detector

text: system.

text: Signiﬁcantly

text: more

text: pions

text: will

text: now

text: tor

text: conﬁgurations.

text: The

text: spectrometers

text: can

text: be

text: rotated

text: around

text: the

text: survive

text: to

text: the

text: detector

text: system

text: for

text: the

text: same

text: kinetic

text: energy,

text: but

text: target

text: chamber

text: in

text: the

text: center

text: and

text: they

text: can

text: be

text: operated

text: in

text: a

text: sin-

text: the

text: data

text: still

text: has

text: to

text: be

text: corrected

text: for

text: pion

text: decay.

text: At

text: the

text: same

text: gle,

text: double,

text: and

text: triple

text: coincidence

text: mode.

text: Looking

text: along

text: the

text: time,

text: muon

text: contamination

text: of

text: the

text: data

text: will

text: be

text: reduced,

text: so

text: that

text: it

text: beam

text: direction,

text: spectrometers

text: A

text: and

text: C

text: are

text: placed

text: on

text: the

text: left

text: is

text: feasible

text: to

text: determine

text: it

text: from

text: a

text: simulation

text: and

text: correct

text: for

text: it.

text: and

text: right

text: side

text: of

text: the

text: beam

text: pipe,

text: respectively.

text: Spectrometer

text: B

text: Therefore,

text: a

text: new

text: short

text: orbit

text: spectrometer

text: (SOS)

text: was

text: con-

text: can

text: be

text: moved

text: to

text: either

text: side

text: of

text: the

text: beam

text: pipe;

text: in

text: standard

text: con-

text: structed

text: speciﬁcally

text: for

text: detection

text: of

text: low

text: energy

text: pions.

text: The

text: ﬁhagvueraatiroenlaittivies

text: (ﬁer,set

text: aSsOeSv.aAlus-

text: ferent

text: electron

text: scattering

text: experiments

text: for

text: detection

text: of

text: charged

text: ated,

text: which

text: at

text: the

text: same

text: time

text: allowed

text: to

text: test

text: the

text: performance

text: of

text: the

text: SOS

text: in

text: detection

text: of

text: low

text: energy

text: pions.

text: Section

text: of

text: this

text: paper

text: describes

text: the

text: magnet,

text: detector

text: sys-

text: Corresponding

text: author,

text: Email:

text: friscic@mit.edu

text: tem

text: and

text: electronics

text: of

text: the

text: SOS.

text: In

text: section

text: the

text: measurement

text: address:

text: MIT-LNS,

text: Cambridge

text: MA,

text: USA

text: used

text: for

text: calibration

text: of

text: the

text: SOS

text: is

text: discussed.

text: Section

text: deals

text: paper

text: comprises

text: parts

text: of

text: the

text: doctoral

text: theses

text: of

text: D.

text: Baumann,

text: M.

text: Ding

text: and

text: I.

text: Frisˇcˇic´.

text: with

text: a

text: measurement

text: of

text: the

text: p(e,

text: e

text: π+)n

text: reaction.

text: A

text: summary

text: and

text: Preprint

text: submitted

text: to

text: Nucl.

text: Instr.

text: and

text: Meth.

text: A

text: May

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 1: Contributions to eﬀective thickness of SOS tracking detector, x is thickness of a layer and X0 is radiation length of the employed material. − Material x X0 (cm) x/(10 6X0) Comment He + C H 152 mm 63900 238 Counting gas 2 6 PET (4+4+12) µm 28.7 69.7 Chamber foils (CF) · Al 2 (10+10+40) nm 8.9 1.35 Coating on the CF Al 367 nm 8.9 4.1 Potential wires (averaged) Drift chambers 313 Sum for the chambers Polyimid 50 µm 28.6 175 Vacuum exit Aramid 6 µm 28.6 21.0 He-ﬂange foil Helium 20 mm 568000 3.5 Gap below the chamber Rest 277 Everything outside the chamber Altogether 590 From target to scintillator are alternating shifted by 100 µm with respect to the center of the cell. In the trajectory reconstruction procedure the shifts are taken into account to determine at which side the particle has passed the signal wire – so-called left-right decision. The signal wires are grounded by the preampliﬁer. The potential wires (cathode wires) at the edges of the drift cells are set to − a high negative voltage, typically 5400 V. In order to obtain drift cells with well deﬁned borders and to mutually suppress the cross talk between neighboring drift cells, potential wires closest to the signal wires are set to an intermediate voltage by a voltage degrader. The best averaged single wire eﬃciency of 95% was achieved, when the ratio of the cathode wire and in- termediate voltage was 4 : 1 [? ]. In order to assure that the boundary drift cells also have a proper geometry of the elec- tric ﬁeld, only potential wires at the edges of the signal wires vertical arrangement are grounded. Figure 3: (colour online) The SOS drift chambers in a build-in position. The bottom chamber is used for reconstruction of non-dispersive coordinates, and the upper chamber for dispersive coordinates. Positions of the signal wires Table 2: Properties of the SOS drift chambers. are marked with white points. Edge rays were determined with help of RAY- TRACE simulation. Outer dimensions 410 x 210 x 156 mm3 Active surface 294.4 x 73.6 mm2 Wire drift cell length x width 18.4 x 5.08 mm2 ± Signal wires horizontal oﬀset 100 µm Number of signal wires 64/16 (X-/Y-chamber) Number of potential wires 355/103 (X-/Y-chamber) Signal wires: (cid:31)15 µm Au-coated tungsten-rhenium Potential wires: (cid:31)80 µm Ag-coated aluminium Gas: 50% helium + 50% ethan The Y-chamber with two drift cells is the ﬁrst chamber above the magnet. It measures the non-dispersive coordinate and an- Figure 4: (colour online) Lateral proﬁle of a single SOS drift cell. Circles repre- gle (ych, φch). The second chamber consisting of eight cells is sent potential wires and crosses represent signal wires, signal wires, which are the X-chamber. The wires of the X-chamber are perpendicular staggered left and right by 100 µm. In order to make the signal wire staggering to the wires of the Y-chamber and, therefore, it measures the visible on this ﬁgure, it is exaggerated. dispersive coordinate and angle (xch, θch). Particles enter the X-chamber after the Y-chamber and due to multiple scattering, resolution of the X-chamber will always be lower relative to Fig. ?? exemplary shows a single SOS drift cell. Each drift the Y-chamber. The priority to Y-chamber was given to pro- cell is made of eight vertically arranged signal wires, which vide better discrimination of particles which may be scattered 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Table

text: Contributions

text: to

text: eﬀective

text: thickness

text: of

text: SOS

text: tracking

text: detector,

text: x

text: is

text: thickness

text: of

text: a

text: layer

text: and

text: is

text: radiation

text: length

text: of

text: the

text: employed

text: material.

font: UZWYCV+txsy

text: Material

text: x

text: (cm)

text: Comment

text: He

font: OOTUFF+rtxr

text: C

text: H

text: mm

text: Counting

text: gas

text: PET

text: µm

font: RCJULN+rtxmi

text: Chamber

text: foils

text: (CF)

text: Al

text: nm

text: Coating

text: on

text: the

text: CF

text: Al

text: nm

text: Potential

text: wires

text: (averaged)

text: Drift

text: chambers

text: Sum

text: for

text: the

text: chambers

text: Polyimid

text: µm

text: Vacuum

text: exit

text: Aramid

text: µm

text: He-ﬂange

text: foil

text: Helium

text: mm

text: Gap

text: below

text: the

text: chamber

text: Rest

text: Everything

text: outside

text: the

text: chamber

text: Altogether

text: From

text: target

text: to

text: scintillator

text: are

text: alternating

text: shifted

text: by

text: µm

text: with

text: respect

text: to

text: the

text: center

text: of

text: the

text: cell.

text: In

text: the

text: trajectory

text: reconstruction

text: procedure

text: the

text: shifts

text: are

text: taken

text: into

text: account

text: to

text: determine

text: at

text: which

text: side

text: the

text: particle

text: has

text: passed

text: the

text: signal

text: wire

text: so-called

text: left-right

text: decision.

text: The

text: signal

text: wires

text: are

text: grounded

text: by

text: the

text: preampliﬁer.

text: The

text: potential

text: wires

text: (cathode

text: wires)

text: at

text: the

text: edges

text: of

text: the

text: drift

text: cells

text: are

text: set

text: to

text: a

text: high

text: negative

text: voltage,

text: typically

text: V.

text: In

text: order

text: to

text: obtain

text: drift

text: cells

text: with

text: well

text: deﬁned

text: borders

text: and

text: to

text: mutually

text: suppress

text: the

text: cross

text: talk

text: between

text: neighboring

text: drift

text: cells,

text: potential

text: wires

text: closest

text: to

text: the

text: signal

text: wires

text: are

text: set

text: to

text: an

text: intermediate

text: voltage

text: by

text: a

text: voltage

text: degrader.

text: The

text: best

text: averaged

text: single

text: wire

text: eﬃciency

text: of

text: was

text: achieved,

text: when

text: the

text: ratio

text: of

text: the

text: cathode

text: wire

text: and

text: in-

text: termediate

text: voltage

text: was

text: In

text: order

text: to

text: assure

text: that

text: the

text: boundary

text: drift

text: cells

text: also

text: have

text: a

text: proper

text: geometry

text: of

text: the

text: elec-

text: tric

text: ﬁeld,

text: only

text: potential

text: wires

text: at

text: the

text: edges

text: of

text: the

text: signal

text: wires

text: vertical

text: arrangement

text: are

text: grounded.

text: Figure

text: (colour

text: online)

text: The

text: SOS

text: drift

text: chambers

text: in

text: a

text: build-in

text: position.

text: The

text: bottom

text: chamber

text: is

text: used

text: for

text: reconstruction

text: of

text: non-dispersive

text: coordinates,

text: and

text: the

text: upper

text: chamber

text: for

text: dispersive

text: coordinates.

text: Positions

text: of

text: the

text: signal

text: wires

text: Table

text: Properties

text: of

text: the

text: SOS

text: drift

text: chambers.

text: are

text: marked

text: with

text: white

text: points.

text: Edge

text: rays

text: were

text: determined

text: with

text: help

text: of

text: RAY-

text: TRACE

text: simulation.

text: Outer

text: dimensions

text: x

text: x

text: Active

text: surface

text: x

text: Wire

text: drift

text: cell

text: length

text: x

text: width

text: x

text: Signal

text: wires

text: horizontal

text: oﬀset

text: µm

text: Number

text: of

text: signal

text: wires

text: (X-/Y-chamber)

text: Number

text: of

text: potential

text: wires

text: (X-/Y-chamber)

text: Signal

text: wires:

text: µm

text: Au-coated

text: tungsten-rhenium

text: Potential

text: wires:

text: µm

text: Ag-coated

text: aluminium

text: Gas:

text: helium

text: ethan

text: The

text: Y-chamber

text: with

text: two

text: drift

text: cells

text: is

text: the

text: ﬁrst

text: chamber

text: above

text: the

text: magnet.

text: It

text: measures

text: the

text: non-dispersive

text: coordinate

text: and

text: an-

text: Figure

text: (colour

text: online)

text: Lateral

text: proﬁle

text: of

text: a

text: single

text: SOS

text: drift

text: cell.

text: Circles

text: repre-

text: gle

text: (ych,

text: φch).

text: The

text: second

text: chamber

text: consisting

text: of

text: eight

text: cells

text: is

text: sent

text: potential

text: wires

text: and

text: crosses

text: represent

text: signal

text: wires,

text: signal

text: wires,

text: which

text: are

text: the

text: X-chamber.

text: The

text: wires

text: of

text: the

text: X-chamber

text: are

text: perpendicular

text: staggered

text: left

text: and

text: right

text: by

text: µm.

text: In

text: order

text: to

text: make

text: the

text: signal

text: wire

text: staggering

text: to

text: the

text: wires

text: of

text: the

text: Y-chamber

text: and,

text: therefore,

text: it

text: measures

text: the

text: visible

text: on

text: this

text: ﬁgure,

text: it

text: is

text: exaggerated.

text: dispersive

text: coordinate

text: and

text: angle

text: (xch,

text: θch).

text: Particles

text: enter

text: the

text: X-chamber

text: after

text: the

text: Y-chamber

text: and

text: due

text: to

text: multiple

text: scattering,

text: resolution

text: of

text: the

text: X-chamber

text: will

text: always

text: be

text: lower

text: relative

text: to

text: Fig.

text: exemplary

text: shows

text: a

text: single

text: SOS

text: drift

text: cell.

text: Each

text: drift

text: the

text: Y-chamber.

text: The

text: priority

text: to

text: Y-chamber

text: was

text: given

text: to

text: pro-

text: cell

text: is

text: made

text: of

text: eight

text: vertically

text: arranged

text: signal

text: wires,

text: which

text: vide

text: better

text: discrimination

text: of

text: particles

text: which

text: may

text: be

text: scattered

text: ##LTFigure##

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