Partially quenched chiral perturbation theory for N=1 supersymmetric Yang-Mills theory
aa r X i v : . [ h e p - l a t ] N ov Partially quenched chiral perturbation theory for N = supersymmetric Yang-Mills theory Gernot Münster ∗ , Hendrik Stüwe Universität Münster, Institut für Theoretische Physik,Wilhelm-Klemm-Str. 9, D-48149 Münster, GermanyE-mail: [email protected]
Adding a gluino mass term to N = The 32nd International Symposium on Lattice Field Theory23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
QChPT for N = supersymmetric Yang-Mills theory Gernot Münster
1. The Model N = SUSY Yang-Mills Theory
The N = A a m ( x ) , a = , . . . , N c −
1, describing gluons belonging to the gauge group SU( N c ), a fermionic spinorfield l a ( x ) , obeying the Majorana condition ¯ l = l T C , and an auxiliary field. The Majorana fielddescribes gluinos, the superpartners of gluons, and it transforms under the adjoint representationof the gauge group: D m l a = ¶ m l a + g f abc A b m l c . The on-shell Lagrangean of SYM in Euclideanspace-time reads L = F a mn F a mn +
12 ¯ l a g m ( D m l ) a . (1.1)It is invariant under the SUSY transformations d A a m = − l a g m e , dl a = − s mn F a mn e . (1.2)Being part of the supersymmetrically extended Standard Model, SYM represents an interestingfield theory. It has some similarities to QCD, the important differences being that gluinos areMajorana particles, and that they are in the adjoint representation, in contrast to quarks.The Lagrangean can be extended to include a gluino mass term m ˜ g ¯ l a l a . The gluino massbreaks SUSY softly. The action is only invariant under supersymmetry transformations in the limit m ˜ g = SYM is an interesting laboratory for the study of the properties of supersymmetric models. Asin the case of QCD, SYM is characterised by a number of non-perturbative aspects, which can beinvestigated in a lattice-discretised version: • SYM has a discrete chiral symmetry Z N c , which is predicted to be broken spontaneouslydown to Z . The breaking is associated with a gluino condensate < ll > = • SYM is expected to show confinement, and the particle states should be bound states, form-ing supermultiplets. • Static quarks, belonging to the fundamental representation of the gauge group, are expectedto be confined. • Spontaneous breaking of SUSY is predicted not to occur for SYM. • SUSY is broken by the lattice regularisation. A question which is still open is if there is acontinuum limit in which SUSY is restored? • Predictions about the low-lying particle spectrum from effective Lagrangeans [1, 2] shouldbe checked on the lattice. 2
QChPT for N = supersymmetric Yang-Mills theory Gernot Münster
Lattice discretisation generically breaks SUSY [3]. In the case of SYM a fine-tuning of thebare gluino mass parameter in the continuum limit is enough to approach both the (spontaneouslybroken) chiral symmetry and supersymmetry of the continuum theory [4, 5]. Based on this, Curciand Veneziano [4] proposed to use the Wilson action and to search for a supersymmetric continuumlimit by an appropriate tuning of parameters. The Wilson action for SYM is given by S = − b N c (cid:229) p Re Tr U p + (cid:229) x ( ¯ l ax l ax − k (cid:229) m = h ¯ l ax + ˆ m V ab , x m ( + g m ) l bx + ¯ l ax V tab , x m ( − g m ) l bx + ˆ m i) , (1.3)where V ab , x m are the link variables in the adjoint representation. The parameters in the lattice actionare the inverse gauge coupling b = N c / g and the hopping parameter k = / ( m + ) , related tothe bare gluino mass m .Numerical simulations of this model, with gauge group SU(2), have been performed by theMünster-DESY-Frankfurt group in recent years, see the contributions to this conference by P. Giu-dice and S. Piemonte, and Refs. [6, 7].
2. The Goals
As a function of the hopping parameter k the gluino condensate makes a jump at a certainvalue k c . In the phase diagram the line k = k c ( b ) represents a first order phase transition.The recovery of both supersymmetry and chiral symmetry in the continuum limit requires totune the hopping parameter to the point k c ( b ) , where the renormalised gluino mass vanishes [4, 5]. The gluino mass term is not protected against additive renormalisation in the Curci-Venezianoformulation. Therefore the point of vanishing gluino mass is not given a priori, but has to bedetermined with suitable observables. A numerically relatively cheap and therefore attractive wayto tune to k c is to search for the point where the adjoint pion mass vanishes: m a– p → p ?A pseudoscalar mesonic bound state, called a– h ′ , is represented by the interpolating field¯ lg l . Its correlator has connected and disconnected pieces: ① ② - 2 ① ② QChPT for N = supersymmetric Yang-Mills theory Gernot Münster
Figure 1:
The phase diagram of SYM with gauge group SU(2)
The correlator of the a– p is now given by the connected part of the a– h ′ correlator, and the adjointpion mass can be obtained unambiguously from it. The a– p correlator has the form of the correlatorof a meson formed out of different gluino species. But since SYM only contains one gluino, thea– p is not a physical particle in the Hilbert space of the theory.The assumption underlying the tuning of k is that the adjoint pion mass vanishes with therenormalised gluino mass as m p (cid:181) m ˜ g , (2.1)analogously to the Gell-Mann-Oakes-Renner (GOR) relation of QCD [8], m p (cid:181) m q . (2.2)An argument for this relation, based on the OZI-approximation of SYM, has been given in [1].On the other hand, the renormalised gluino mass m ˜ g can be determined by means of the latticesupersymmetric Ward identities as discussed in [9]. Numerical investigations of both m ˜ g from su-persymmetric Ward identities and m a– p have been performed in [10]. The results are in agreementwith am ˜ g Z S = (cid:18) k − k c (cid:19) , ( am a– p ) ≃ A (cid:18) k − k c (cid:19) , (2.3)and support the validity of the above assumption, see Fig. 2. The a– p , however, yields a moreprecise signal for the tuning than the supersymmetric Ward identities. The goals of this work are: 4
QChPT for N = supersymmetric Yang-Mills theory Gernot Münster • define the adjoint pion a– p properly, • establish the relation m p (cid:181) m ˜ g . In QCD the GOR relation can be derived in the framework of chiral perturbation theory. Thusthe idea is to use this as a starting point for SYM, too. In contrast to QCD, however, SYM does nothave a continuous chiral symmetry. Therefore the approach consists in adding additional flavoursof gluinos, l i ( x ) , i = , . . . , N , which are quenched, in order to keep SUSY intact. This is a par-ticular case of Partially Quenched Chiral Perturbation Theory (PQChPT), in the spirit of the caseof one-flavour QCD [11]. With the help of the additional gluinos, adjoint pions can be formed as¯ l i g ( t a ) i j l j with i , j = ,
3. The Calculation
Let us start by extending SYM with N − l i ( x ) , i = , . . . , N . Incontrast to QCD, where the chiral symmetry group of N quarks is given by SU ( N ) L ⊗ SU ( N ) R , dueto the Majorana condition the chiral symmetry group of extended SYM turns out to be given by asubgroup isomorphic to SU( N ). If the gluinos are represented as Weyl fermions, this SU( N ) is thegroup of transformations of N Weyl fermions.Spontaneous breakdown of chiral symmetry, accompanied by non-vanishing gluino conden-sates, breaks the group G = SU ( N ) down to H = SO ( N ) . To be specific, we consider the case N = G / H = SU ( ) / SO ( ) ∼ S . It canbe parameterised by u = exp ( i a T + i a T ) , where T i are the generators of SU(2). It is now conve-nient to define the nonlinear Goldstone boson field by U ( x ) = u ( x ) = u ( x ) u ( x ) T . = exp ( i f ( x ) / F ) , ,!(cid:4)(cid:11)(cid:6)(cid:28)(cid:9) (cid:9)(cid:12)(cid:13)(cid:12)(cid:14)(cid:9)(cid:28)(cid:4)((cid:4)(cid:3) (cid:9)(cid:24)(cid:9)!(cid:14) (cid:1)8(cid:23)1(cid:10)(cid:17)%(cid:2)(cid:21)(cid:7)(cid:6)(cid:3)& (cid:31)(cid:6)0(cid:19)(cid:16)(cid:15)(cid:16)(cid:10)(cid:2)(cid:5)(cid:6)(cid:11)(cid:8)’’(cid:16)(cid:13)5(cid:6)’(cid:2)(cid:19)(cid:2)(cid:3)(cid:4)(cid:15)(cid:4)(cid:19)(cid:28) Figure 2:
The renormalised gluino mass from SUSY Ward identities (left figure) and the adjoint pion masssquared (right figure) as functions of k QChPT for N = supersymmetric Yang-Mills theory Gernot Münster because the transformation law of this group valued field, U ( x ) → U ′ ( x ) = VU ( x ) V T , V ∈ SU ( ) , (3.1)is similar to that of the usual chiral perturbation theory.Analogously to the approach used in QCD, the leading order effective Lagrangean can bedetermined to be L = F ( ¶ m U ¶ m U † ) + F ( c U † + U c † ) , (3.2)where c = B m ˜ g is the symmetry breaking mass term, and F and B are low-energy constants.Note that the theory with 2 gluinos might be conformal or near-conformal [12], implying adifferent breaking pattern. However, its discussion here just serves as a preparation of the followingPQChPT analysis, which is not affected by this possibility. In order that the dynamical content of the model is identical to that of SYM, and the correlationfunctions of the original fields are unchanged, the additional gluinos have to be quenched, whichmeans that they are not taken into account in the fermionic functional integral. This is a case ofPQChPT [13, 14]. Partial quenching can be described theoretically by the introduction of bosonicghost fermions [15], in our case ghost gluinos. The contribution of the ghost gluinos exactly cancelsthe contribution of the additional gluinos, and only the contribution of the original single gluinoremains. In the case of N = r ( x ) , compensating the contributionof the additional gluino. The resulting chiral symmetry group is the graded group SU(2 | f = f ss f sv f sg f vs f vv f vg f gs f gv f gg , (3.3)where s , v and g stand for sea, valence and ghost. Now the adjoint pion can in this formulation bedefined to be the meson represented by f sv .The leading order effective Lagrangean for the partially quenched theory is given by L PQ = F ( ¶ m U ¶ m U † ) + F ( c U † + U c † ) , (3.4)where str denotes the supertrace. The next-to-leading order terms can be constructed analogouslyto the NLO terms for QCD [16], and are not reproduced here. They contain further low-energyconstants L i , the so-called Gasser-Leutwyler coefficients.The masses of the pseudo-Goldstone bosons can be calculated in PQChPT by means of theeffective Lagrangean. We have calculated them in NLO. Whereas the tree-level contributions aresimilar to the ones in QCD, the loop contributions differ due to the different group structure. Forthe mass of the adjoint pion m a– p we find m p = B m ˜ g + ( B m ˜ g ) F ( L − L − L + L ) , (3.5)6 QChPT for N = supersymmetric Yang-Mills theory Gernot Münster with the low-energy coefficients L i mentioned above. For small m ˜ g we recognise the desired GOR-relation m p = B m ˜ g . (3.6) To summarise, the results of this investigation are: • The adjoint pion a– p is defined in PQChPT, • m p = B m ˜ g in leading order PQChPT.Details can be found in [17]. References [1] G. Veneziano and S. Yankielowicz, Phys. Lett.
B 113 (1982) 231.[2] G. R. Farrar, G. Gabadadze and M. Schwetz, Phys. Rev.
D 58 (1998) 015009[arXiv: hep-th/9711166 ].[3] G. Bergner, JHEP (2010) 024 [arXiv: [ hep-lat ]].[4] G. Curci and G. Veneziano, Nucl. Phys. B 292 (1987) 555.[5] H. Suzuki, Nucl. Phys.
B 861 (2012) 290 [arXiv: [ hep-lat ]].[6] G. Bergner, I. Montvay, G. Münster, U. D. Özugurel and D. Sandbrink, JHEP (2013) 061[arXiv: [ hep-lat ]].[7] G. Bergner, I. Montvay, G. Münster, U. D. Özugurel and D. Sandbrink, PoS(LATTICE 2013) (2013)483 [arXiv: [ hep-lat ]].[8] M. Gell-Mann, R. Oakes and B. Renner, Phys. Rev. (1968) 2195.[9] F. Farchioni, A. Feo, T. Galla, C. Gebert, R. Kirchner, I. Montvay, G. Münster and A. Vladikas, Eur.Phys. J. C 23 (2002) 719 [arXiv: hep-lat/0111008 ].[10] K. Demmouche, F. Farchioni, A. Ferling, I. Montvay, G. Münster, E. E. Scholz and J. Wuilloud, Eur.Phys. J.
C 69 (2010) 147 [arXiv: [ hep-lat ]].[11] F. Farchioni, I. Montvay, G. Münster, E. E. Scholz, T. Sudmann and J. Wuilloud, Eur. Phys. J. C 52 (2007) 305 [arXiv: [ hep-lat ]].[12] A. Athenodorou, E. Bennett, G. Bergner, B. Lucini and A. Patella, PoS(LATTICE 2013) (2013) 066[arXiv: [ hep-lat ]].[13] C. W. Bernard and M. F. L. Golterman, Phys. Rev. D 49 (1994) 486 [arXiv: hep-lat/9306005 ].[14] S. R. Sharpe, Phys. Rev.
D 56 (1997) 7052 [Erratum-ibid.
D 62 (2000) 099901][arXiv: hep-lat/9707018 ].[15] A. Morel, J. Phys. (France) (1987) 1111.[16] J. Gasser and H. Leutwyler, Ann. Phys. (1984) 142.[17] G. Münster and H. Stüwe, JHEP (2014) 034 [arXiv: [ hep-th ]].]].