A low temperature expansion for matrix quantum mechanics

Journal of High Energy Physics, May 2015

We analyze solutions to loop-truncated Schwinger-Dyson equations in massless \( \mathcal{N}=2 \) and \( \mathcal{N}=4 \) Wess-Zumino matrix quantum mechanics at finite temperature, where conventional perturbation theory breaks down due to IR divergences. We find a rather intricate low temperature expansion that involves fractional power scaling in the temperature, based on a consistent “soft collinear” approximation. We conjecture that at least in the \( \mathcal{N}=4 \) matrix quantum mechanics, such scaling behavior holds to all perturbative orders in the 1/N expansion. We discuss some preliminary results in analyzing the gauged supersymmetric quantum mechanics using Schwinger-Dyson equations, and comment on the connection to metastable microstates of black holes in the holographic dual of BFSS matrix quantum mechanics.

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A low temperature expansion for matrix quantum mechanics

Received: October A low temperature expansion for matrix quantum Ying-Hsuan Lin 0 1 3 4 Shu-Heng Shao 0 1 2 4 Yifan Wang 0 1 2 4 Xi Yin 0 1 3 4 0 77 Massachusetts Ave. , Cambridge, MA 02139 , U.S.A 1 17 Oxford St. , Cambridge, MA 02138 , U.S.A 2 Center for Theoretical Physics, Massachusetts Institute of Technology 3 Jefferson Physical Laboratory, Harvard University 4 Open Access , c The Authors We analyze solutions to loop-truncated Schwinger-Dyson equations in massless N = 2 and N = 4 Wess-Zumino matrix quantum mechanics at finite temperature, where conventional perturbation theory breaks down due to IR divergences. We find a rather intricate low temperature expansion that involves fractional power scaling in the temperature, based on a consistent soft collinear approximation. We conjecture that at least in the N = 4 matrix quantum mechanics, such scaling behavior holds to all perturbative orders in the 1/N expansion. We discuss some preliminary results in analyzing the gauged supersymmetric quantum mechanics using Schwinger-Dyson equations, and comment on the connection to metastable microstates of black holes in the holographic dual of BFSS matrix quantum mechanics. matrix; quantum; Supersymmetric gauge theory; Gauge-gravity correspondence; M(atrix) The- - mechanics Contents 1 Introduction 2 The action 2.2 Schwinger-Dyson equations N = 2 SUSY Ward identities 3 The low temperature limit A soft-collinear approximation Continuum limit Free energy Higher loop corrections Action, Schwinger-Dyson equations, and free energy 5 Towards the low temperature expansion of BFSS Higher loops N = 2 gauge-fixed BFSS The N = 2 ghost determinant 5.3 The free energy N = 4 gauge-fixing 5.5 Phases of BFSS 6 The high temperature limit Schwinger-Dyson equations The free energy 6.3 Size of the wave function 7 Further discussions 7.1 More supersymmetric gauges mensions A Hawking decay rate of D0-branes A.1 Greybody factor A.2 Partial wave summation A.3 Decay rate B.1 Superspace B.2 Vector multiplet Connections Supersymmetry transformations Field strengths B.3 Matter multiplet C 1D N = 4 SUSY C.1 Superspace and convention C.2 Chiral multiplet C.3 Vector multiplet N = 4 SUSY Ward identities D Computation of xn Computation of the free energy BFSS G.1 Subleading corrections to the self-energies G.2 Free energy in the high temperature limit G High temperature analysis of one-loop truncated N = 2 gauge-fixed Introduction limits where the Monte Carlo computation becomes costly. supersymmetries. N units of D0-brane charge [25] C1 = f 1 f c0gsN ls7 1 + A c0 = 603, A = 1 r7 0 The Hawking temperature of the black hole is TH = (gY2MN )1/2 (gY2MN )1/3 low temperature limit. infinity (exponentially) in the infinity N limit. probe D0-brane in the background (1.1) SD0 = TD0 dt f 1A1/2p1 + f A2r2 dt f 1 1 + A namically stable. The latter is expected when r0 the decay rate is exponentially suppressed. N 1/9lP . This is also the regime where using Schwinger-Dyson equations. This is the approach of [2628], where the authors T /(gY2MN )1/3 results in the N the cubic superpotential W We refer to such a theory as an of the self-energies of all other modes. hr = sign(r) sign(n)Cn + sign(n)Cn n sign(r)Cr + O(1), where Cn = P 7/5 + O(2). mean-field approximation [26]. The result is F = const 2 3 N 2Nf 6/5 + O(9/5). free energy interpolates between T 9/5 and T 6/5. and Model II, the N is left for the future. = 4 Wess-Zumino matrix theory is analyzed in appendix G. The action real superfields1 interacting through a cubic superpotential a = a + ia + if a2, W = 6c After integrating out the fermionic coordiantes, the action is3 S = 1The i in front of f gives the right sign for the kinetic term of f . {Q, } = i f + i, [Q, f ] = i . Schwinger-Dyson equations non-perturbatively. We will make a equations, and then discuss the validity of such truncations. f a = nZ rZ+ 12 nZ S = Tr 1 X 2ir a 1 X fanfn X abcfankbnck + X abc ars,r,s . b c Let us denote the exact propagators by hanbmi nabn,m, h,r,si igrabr,s, a b = 1 + 1 X knk, 2 X kgrk. 2 X k nk + 2 X grgnr, tional perturbation theory. n ( 2n )2 + n written in terms of the self-energies, are hr = 2 X 2 X h( 2k )2 + ki [1 + nk] 2 X 10 the fermion propagator gr, respectively. to the Schwinger-Dyson equations (2.12). continuous, we write the exact two-point functions (2.8) as If SUSY is not spontaneously broken, then In momentum space, (2.14) and (2.15) read hf (p)f (p)i = p + h(p) 11 hr = s 1 The low temperature limit A soft-collinear approximation numbers n, r are assumed to be O(1)) modes as hard and zero modes as soft. 12 hr = + Cr, According to the scaling ansatz (3.1), X k nk + X grgnr , k6=0 k6=0,n 2 X kgrk, 1 X kk k6=0 k6=0 0 1 + n 4/5, 2n Bn 3/5, hr = sign(r) 13 sn(1 + xn), 2n sn(1 + xn)1, sn sign(n) directly from the Schwinger-Dyson equations, and find xn = + O(6/5) 3/5, in accordance with (2.16). n6=0 s2n in (3.2), and we find 1 X n6=0 42n2 s2n(1 + xn)2 + O(3/5) = For sn and hr, let us write sn = sign(n) hr = sign(r) + s(n1) + s(n2) + O(8/5), + h(r1) + h(r2) + O(8/5), g , Ck X sign(`)sign(k `) `6=0,k 14 h(r1), and h(r2) can be found in appendix E. The results are s(n1) = h(r1) = Cn + O(1), + sign(n) 22n2 + O(8/5), k6=0,2n 82 Cn2 42 k6=0,n X ()ksign n 2 82 Cr2 42 k6=0 + sign(r) k6=0 k6=0,n X sign(k)sign(r k)xk + O(8/5). from which we obtain X sn2 X hr2 = g n6=0 16 3 304 + O(16/5), 52 57 + O(2). Continuum limit X sign(k)sign(n k)xk 15 of the holonomy matrix along the thermal circle is involved. Free energy truncated Schwinger-Dyson equations: S0 = Tr F = F0 + hS2 S0i0 2 hS32i0. h i0 DDDf eS0 . action, i.e., 1 X ln n 2 n 1 X ln n + 1 X ln gr2, for the nonzero modes can be written as (p) = | | 1 + = | | 1 + k=1 2k |p| + O(2) at x = 0, taking p 0 gives hS2 S0i0 = 1 X 1 X mk m+k + 1 X grgsr+s. 1 X( n 1) gr 1 , hS2 S0i0 = 1 22 |n| n6=0 n6=0 |n| X n X n (0)3 + O(9/5), (0)3 + O(9/5). F = const 4 3 N 2Nf 6/5 + O(9/5), where we have restored the overall N 2Nf factor. We have also convention of [26]). Higher loop corrections the fermion propagator gr, respectively. term6 1 + 3 1 + scaling solution in the low temperature limit. 18 The Yang-Mills action in the WZ gauge is SYM = 2 2 D0 2 i[Xi, ] 2 D0XiD0Xi 4 [Xi, Xj][Xi, Xj] , component fields to be = Xi + 21 i, = A0 i 2 = + 2 2 [A0, ] 2i i[Xi, ] + 2 [d, ] 2 [, ] Matter multiplet transformations are = + i + if 2. = if + i, The super gauge transformation is Written in terms of the component fields, we have The action we will consider is SM = 4 aa and write them in terms of the self-energies as sn = ( 2n )2 + nG vector : matter : ghost : ar = 2n = un = The one-loop truncated Schwinger-Dyson equations are 3 X r ( 2r + rM )( 2(nr) + nMr) r ( 2r + rG)( 2(nr) + nGr) rV = 3 2r X + 2 1 + (2n)2nM , 8 1 2r + rM , 2 X n (1 + (2n)2nM)((2(rn))3 + rVn) 4 X nG = 1 2n 2 X 7 X r ((2r)3 + rV )(2(nr) + nMr) 4 X n (2n + 2nVn)2 F0 = 2 log X log l2 + X log ar 2 l l r 7 X log l2 + 7 X log gr 7 X log l2 + X log sl 2 X log tr + X log ul, l6=0 l2 1 + X 7 X l6=0 tr 1 X (ul 1) , i 2 X 2r 2s aras + 7 X 2 1 X l2m2 + 14 X l2m2 4 X 14 X 7 X 2 X 7 X g 1 X tr + l2trts ultras + 1 X sltras and quartic couplings. 1D N = 4 SUSY 43 Superspace and convention = = (iI, ~). Q = + , D = . The only nontrivial anticommutators are = , = , In particular, = = = = = . = 21 , = = 2 , Chiral multiplet We can solve this constraint by first introducing a(y, ) = a(y) + in the action positive. The SUSY transformations are [Q, a] = i2a, {Q, a} = {Q , } = i2 a, [Q , f a] = a L = Vector multiplet the following component field expansion The gauge transformation is e2V e2i e2V e2i The field-strength superfield is defined by The kinetic action for the vector multiplet is then given by L = Tr fa(p)f b(p)i = a h (p)b(p)i = i p + h(p) Going to the momentum space, this implies that is Similarly, we can consider 0 = h{Q, a( )fb( 0)}i = 2 hf a( )fb( 0)i In the momentum space it is Using = , we have In summary, we have obtained the exact relation hfb(p)f a(p)i = ip b h(p)a(p)i. h(p) = 1 R is set to zero in the zero mode truncation. We will write The field-strength superfield is similarly for D. Using dse2sRAe2(1s)R Z 1 usual Wess-Zumino gauge. = e2R + ds e2sR 2 Ae2(1s)R = e2R + 2 ds e2sR Ae2(1s)R du e2uRAe2(su)RAe2(1s)R, 47 we then expand dse2sRAe2(1s)R Z 1 We can simplify the above equation by using the identity The zero mode terms in the Lagrangian can then be written as L = 3 gY2M dse2sRAe2(1s)R Z 1 ) + ] A+AA+A, k6=0,n nk 1+xnk +X k sksnk 1+xk r hrhnr 2|n|0 1 + O(6/5)i . r 20h1+O(6/5)i We have used Computation of xn (SL)(SL) = 1 ( Using (3.5), (3.7) and (3.8), the equation for xn is 0 2n + sn(1 + xn)1 + An #1 (1 + xn)2 = An = Bn = The Schwinger-Dyson equation for xn can thus be written as 2n (1 + xn)2 1 = An (1 + xn)2Bn + O(11/5), 48 k6=0,n ksksnk 1 + xnk 1 + xk (1 + xn)2 (1 + xk)(1 + xnk) 1 + xnk + X k6=0,n sksnk 1 + xk r hrhnr = (1 + xn)2 1 + O(3/5). xn)2Bn 1. Write sk = sign(k) + s(k1), hr = sign(r) where s (k1), h(r1) are of order 2/5. We have hr = 1 X n6=0 ksk(1 + xk)hrk s(k1) = h(r1) = `6=0,k The equation for xn can be written as (1 + xn)2 1 2 k6=0,n k6=0,n k6=0,n 0 X sign(k)sign(n k) 1 + xnk 1 + xk 1 + xk 1 (1 + xn)2 (1 + xk)(1 + xnk) r 20+O(3/5). X sign(nk)s(k1) X sign(nr)h(r1)+ X sign(n k)s(k1) X sign(n r)h(r1) k6=0,n 4 `6=0 | | k6=0,n,` sign(n k)sign(k `) unless the r.h.s. vanishes to leading order. xn = 0 2 an + O(6/5), k6=0,n 0 = 2n X sign(k)sign(n k) (ank an) + `6=0 | | k6=0,n,` = X 1 `6=0 | | an now obeys the equation sign(n k)sign(k `) X sign(n r)sign(r `) k6=0,n sign(n k)sign(k `) X sign(n r)sign(r `) 4 sign(n)HN (|n| 1) + an = 3 + 2 X sign(k)ak k6=0,n |n k| HN (m) k=1 k an = xn = 22n2 (1+xn)2 1 = `6=0 | | k6=0,n,` Here we used that is, A solution is given by 50 1 X 0 hr 20hr2 + k6=0,n ksksnk k6=0 ksk(1 + xk)hrk 1 X k6=0,n sksnk 0 s2n (An + Bn) + O(8/5), r hrhnr 1 X k6=0,n ksksnk g = (xk + xnk) + O(8/5). sn and hr to the relevant order. sn = hr = An = Bn = An = Bn = + Bn(1 + xn) + O(8/5) We can then write sn = sign(n)g1 + hr = sign(r)g1 + X (1)ksign n 2 X n k X sign(k)sign(n k) k6=0,n (xk + xnk) + O(g8). `6=0,k C k2 k6=0,n 2g6Cn2 k6=0,n 2sign(n) 7 X (1)ksign n 2 k6=0,2n k6=0,n = 1 (ghr)2 = 1 hr = 1 k6=0 g4sn X sign(k)sign(n k) (xk + xnk) k6=0,n The equations for sn and hr are equivalent to (gsn)4 = 1 sn 2g4snCn + g4AnBns2n + g2sn(An + Bn) + O(g9) k6=0,2n + sign(r) 3 X sign(k)sign(r k)xk + O(g9), k6=0,n It follows that X(gsn)2 n6=0 X(ghr)2 = X(1)` 3 X(1)`sign(`)C ` + 2 `6=0 `6=0 X(1)`C2` 52 sign(r) X sign(k)sign(r k) k6=0 kskhrk sign(r)Cr + s(n2) and h(r2) can then be solved to be X ()ksign n 2 3 X sign(k)sign(nk)xk + k6=0 sign(r) X sign(k)sign(r k) g6 22 22 sign(r) X sign(r k)Ck + sign(k)Crk k k6=0,n k6=0 n6=0, k2 n6=0 X(1)`sign(`) X k6=0,n 13 g6 + O(g9) = 1 2 6 where we have used sign(n k2 )sign(n) `6=0 sign( 2` k)Ck + sign(k)C 2` k +sn(1+xn) +Xln +sn(1+xn)1 X n 2Xln (0)3 + O(9/5), X(1)`sign(`) X `6=0 k6=0, 2` n6=0 X()nCn2/2 = 3 sign(k)C 2` k = 3 Computation of the free energy energy for the trial action S0, +Xln = const + g k6=0, 2` k6=0 n6=0 |n| by the terms in 21 hS32i0. 1 To compute 2 hS32i0, it will be convenient to recall our solutions for the propagators, (n0) + (n1) + (n2) + O(1), (n1) 2|n| g gs|(n2|) + 2n + sn(1 + xn)1 (n0) + (n1) + (n2) + O(3), 2 C|n| + gs|(n2|) (1 xn + x2n) + g n 2 C|n| xn , = g sign(r) 1+ 2 C|r| +gh(r2) +g | | gr(0) +gr(1) +gr(2) +O(2), gr(0) g sign(r), g(1) r sign(r) 2 C|r|, gr(2) g sign(r) gh(r2) + | | 42 C|2r| g n = n n Using the above, 1 X `6=0 n6=0,` `6=0 1 X `6=0 n6=0,` `6=0 r n6=0 = g |n| |r|g X n (0)3 +O(9/5) 54 X n n 0 r>0 `6=0 r 1 X X grg`r` 2 X ()k + (1 + 0) 42 n=1 n2 + X (n0) (n1) 0 r>0 X (n0) (n2) 0 r>0 X gr(2)gr(0) 0 r>0 `6=0 n6=0,` X gr(0)g`(0)r(`1) + O(9/5) 1 X `6=0 n6=0,` 2 X X gr(1)g(0) (0) `r ` `6=0 r 3 where we have used Subleading corrections to the self-energies equations (see appendix B.4), with a, b, c, d defined in (6.2), 4 X 4 X 2 (0V )2 = 3 b2 2 X m6=0 3 X 2 (0V )2 (0M )2 = 3 b2 0 0 12 a2(d + 1) 1/2 a(d + 1)2 12 X 13 X m6=0 2m2 = 3 , The solution to these algebraic equations are, (1)M = 2.52, (1)V = 0.32, (1)M = 0.683/2, (1) = 5.53. 0 0 0 Now for nonzero modes, 4 X 4 X + r 2r 2(n r) 2(1) 8(1) 2(n r) 2 + 2 + 8 (0V )2 3 X b2 + r 2r 2(n r) Vn (0V )2 (0M )2 0 0 5 X 7 X (0V )2 (0M )2 3 m6=0 1 X We have used 2(a 6b) r r(n r) = 0, r r3(n r) 3 2n2 a2 + 6 4(a 6b) Therefore, from (6.3) and (G.3), = 5.24183 + = 4.86502 + 1 b2 Free energy in the high temperature limit F (0) = f () f 7 X log l2 + 7log sinh kinetic energy), l6=0 l6=0 7 X M 2 l6=0 l l6=0 l 7 X lM + X l6=0 7 X 2 l6=0 7 X l 2 X r 2r r 2 1 + 0M l 7 X Note that to this order 7(7a + 36b) 3/2(a2 + 6)2 a(a2 + 6) Up to an additive constant, 105b4 2338b2 2220 3/2 Open Access. limit, Nucl. 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Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, Xi Yin. A low temperature expansion for matrix quantum mechanics, Journal of High Energy Physics, 2015, 136, DOI: 10.1007/JHEP05(2015)136