F].SKVIER

UCLEAR PHYSIC5

Nuclear Physics B (Proc. Suppl.) 42 (1995) 493~495

PROCEEDINGS SUPPLEMENTS

Infrared Behaviour of Systems With Goldstone Bosons N.D. Hari Dass ~, H.S Sharatchandra, R. Anishetty and R. Basu Inst i tute of Mathemat ical Sciences, C.I.T Campus, Madras 600 113, INDIA

Various nonperturbative techniques are given for handling the infrared behaviour of systems with Goldstone bosons. These include an explicit form for the generating functional of OPI-vertices of the O(N)-models in the large-N limit, analysis based on Ward identities, as well as a renormalisation group picture. Some consequences are discussed.

1. L a r g e - N A n a l y s i s

Wherever there are Goldstone bosons one should expect drastic modifications of the IR- behaviour,part icularly for 2 < d < 4. For full details see [1]. Consider the model described by

S = / d d x ( ~ ( O u ' ( x ) ) 2 - U ( ' 2 ( x ) - C 2 ) 2) (1)

where ~ transforms as the fundamental rep of O(N). A classical analysis indicates two phases: i)a symmetr ic phase where all particles are degen- erate and massive, and ii)the ordered s tate with N - 1 Goldstone modes and a massive longitudi- nal mode u.

The large-N model is solved exactly with the help of an auxiliary field A. The exact renormalised ef- fective action is given by

i l 1~¢ 2) + - ~ T r , l n V (2) 2

where V = - 0 2 -)~, # is the renormalisation scale and T r ~ means omitt ing the loca l part of the first two terms in the expansion around ;~ = #.For the nonlinear model )~2 would be absent classi- cally, but would be induced quantum mechani- cally.Thus the nonlinear model is renormalisable in this limit only in d = 3. In the ordered phase a- A mixing is a crucial fea- ture. IR-behaviour of various propagators is

i6i j i < ir~Trj > ~ kS < a a >,,~ I k I 4------~'

*Present Address; Theory Division, KEK, Tsukuba, Ibaraki, Japan 305

0920-5632/95/$09.50 1995 Elsevier Science B.V. S S D I 0920-5632(95)00290-1

i < ~ > ~ - < ) ~ > ~ i k 2 (3)

V

The striking result tha t the longitudinM suscep- tibility XL diverges as .~ H d/2-2 is reproduced. Even in d = 4, ~L "~ l n H . The leading IR- behaviour of all correlation functions is contained in

il(& + 2w)(k)l 2] (41 FIR d d k [ ~ + 4 < ac~ >

But all non leading IR-behaviour can also be com- puted from the exact effective action ( 2 ) . All multi-pion scattering amplitudes resulting from this vanish as ~ p2. Thus, despite the singular- ities induced by Goldstone modes, the soft-pion theorems remain valid.

2. W a r d I d e n t i t i e s

A nonperturbat ive analysis of the IR-behaviour for the case of arbi t rary N and d can be made in terms of the Ward identities

[¢i( - H i ) - eJ(~-il - Hi)(x)] -- 0 (5)

when an external field /q is present. Successive differentiations of (5) w.r.t, the fields yields many identities. A particularly useful one is

r~2 (p ) - r(/'~ (p) = < ~ > r , , , ( p , 0, - p ) (6)

As a consequence of the Ward identity (5), the effective potential for constant fields and H has the form,

v(g)= (7)

494 N.D Hari Dass et aL /Nuclear Physics B (Proc. Suppl.) 42 (1995) 493-495

which can be used to study OPI at zero momenta. By a systematic analysis of the identities resulting from this, one establishes that i) as far as leading IR-behaviour is concerned

~ ( S )

and that ii) the ~ - r-scattering amplitude van- ishes as H.These two together yield

XL ~ H d / 2 - 2 2 < d < 4 XL "~ l n H d = 4(9)

(6) then implies that the crlrTr 3-point function vanishes as H2-a/~.This is the other manifesta- tion of the Goldstone mode fluctuations.Again, repeated use of identities resulting from (7) show that all multi-pion scattering amplitudes vanish a sH .

Similar conclusions can be drawn from an anal- ysis of the Schwinger-Dyson eqns of the theory. The proof is somewhat technical and will not be reproduced here.Detials can be found in [1].

3. Renormalisation Group Analysis

A deeper understanding can be obtained by considering the RG flows . At T = T¢ (H = 0), the IR-dynamics is governed by the O(N) non- Gaussian fixed point Af~N when 2 < d < 4. RG-flow is from the IR-unstable O(N) Gaussian fixed point ~v to AfGN. For T > Tc we have the symmetric phase with massive (and degenerate) a and 2. The IR-behaviour is governed by the trivial fixed point T corresponding to zero cor- relation length. For T < To, the flow is towards T = O .

The O(N) non-linear or- model is renormaliz- able and asymptotically free in d=2. In d = 2+e, the fl-function has a non-trivial zero at t = tc in addition to the trivial zero, t = 0, corresponding to the theory of (N-l) free scalars. At t = t~ the theory is scale invariant and corresponds to the critical theory with N massless scalars and with anomalous dimensions ( denoted by AfG~v ). The fixed point Af~N is to be identified with AfG~v of the non-linear model. For t > to, the coupling constant grows with the distance, and as in d = 2, the theory is in the unbroken phase. For t < t~ the IR-flow is towards t = 0. The IR- behaviour is of O(N-1) Gaussian theory (This is denoted by

GN-X ) and the ultraviolet behaviour is of the critical theory with anomalous dimensions.

The fact that the infrared behaviour for t < t~ is governed by the O(N-1) Gaussian fixed point gives exhaustive information on the effects of quantum fluctuations of the Goldstone bosons.It means that the pions decouple from each other at low momenta, and their propagators behave as the free propagators for small momenta.It also al- lows us to obtain the non-leading IR-behaviour as well as the effects of an external magnetic field.In particular we now understand why various heuris- tic considerations all give the correct answer: it is the fortuitious behaviour of a free massless the- ory.

The case d=4 is different because of the absence of A/GN as borne out, for example, by the 1 I N expansion. But even in d=4 the pion dynamics is governed by ~g-a . The a- propagator has an uni- versal logarithmic IR-singularity. This feature is present whether a-propagator has a pole at some k S ¢ 0 or not, which is a non-universal feature depending on the microscopic dynamics .

4. S o m e C o n s e q u e n c e s

4.1. QCD and Ferromagnets There are interesting consequences of this phe-

nomena both for finite temperature QCD as well as QCD at zero temperature. In the case of the former, under the well argued scenario that this is a second order transition, WUczek and Rajagopal [2] conclude that the transition must belong to the universality class of d = 3 0(4) magnet models. In numerical simulations of the QCD chiral phase transition at finite temperature, the quantity that has been used to characterise the nature of the phase transition is the so-called A-cumulant de- fined by [3]

0 l n ¢ ¢ A _ - - (lO) Olnmq

where ¢ ¢ is the chiral condensate and mq the current-quark mass. Thus

T > T c ! b ¢ ' ~ m q A = I

~I6 A = 1/,~ (11) T ---- Tc ~ b ,.,., mq

ND. Hari Dass et al./Nuclear Physics B (Proc. Suppl.) 42 (1995) 493-495 495

Below To,but close to it, one should expect, 1,'1 T < T~ ~ --- mq'" (12)

The same considerations apply to the sponta- neously broken phase of zero temperature QCD. Here one should find

A ~ m f l n m q (13)

characterstic of Goldstone phenomena in four di- mensions. The fact that heuristic one-loop arguments also give these results, which are exact, is, as has been stressed before, the consequence of the fixed point dynamics being that of a free massless theory.

Of course, the arguments that map the prob- lem to a classical stat mech problem in d = 3 is valid exactly at Tc [21. But continuity would de- mand that at temperatures in the vicinity of Tc too the effective dimensionahty should be close to 3. Thus, the expectations based on (11- 13 ) are the correct ones. This should yield the ex- perimentally interesting signal tha t for QCD at finite temperatures T < To, XL should smoothly go over from (12) to (13) as T is lowered. Indeed, wherever there are Goldstone bosons, one should expect to see the singular behaviour of XL. It is therefore somewhat disappointing that clear experimental signatures of this effect have not yet been found even for the classic ferromagnets. The lack of isotropy in most real life ferromag- nets would make the experimental establishment of this effect a challenging one. The fact that the behaviour of XL is 'super-universal' in the sense that it depends only on dimension and the exis- tence of Goldstone modes should be established in as many diverse experimental circumstances as possible. The effect should also be observable in the superfluid phase of He. Numerical simula- tions of the O ( N ) - models in three dimensions should also establish this effect.

4.2. P i o n - N u c l e o n I n t e r a c t i o n s Pion- Nucleon interactions can be introduced

into the formalism by adding

= g (a + (14)

It is important to ensure that the singular be- haviour of the a-propagator does not spoil the low

energy theorems for the S-wave lrN-scattering amplitude. Tree level derivation of this phe- nomenologically succesful result is based on a partial cancellation between the a-mediated am- plitude and the pion 'compton'-like contribu- tions.To see that the singular behaviour of the a-propagator does not change the conclusion, no- tice that in our auxiliary field formalism this con- tribution is actually mediated by the A - a mixed propagator as a has no direct interaction with the pions. This propagator is not singular ( 3 ) and the usual cancellation with compton-like contri- butions is still valid.

4.3. Ch i r a l P e r t u r b a t i o n T h e o r y As the IR-behaviour is governed by the flow to-

wards ~¢-1 , one can study the non-leading cor- rections to the infrared behaviour by parametris- ing the microscopic details in a few irrelevant and marginal operators around GN-a and computing their effects. This approach provides a techni- cally more efficient and conceptually simpler al- ternative to chiral perturbat ion theory, which fo- cusses on the infrared divergence of the Goldstone bosons instead of the non-renormalizable ultravi- olet divergences. Our large-N technique also al- lows explicit computation of the correlation func- tions even far away from the infrared region. This provides a useful non-perturbative technique for obtaining a qualitative understanding of the re- sults of chiral perturbation theory. These remarks will be elaborated elsewhere.

R E F E R E N C E S

1. R. Anishetty, R. Basu, N.D. Hari Dass and H.S. Sharatchandra, IMSc Preprint 94-52.

2. F.Wilczek, Int.J.Mod.Phys.AT(1992) 3911; K.Rajagopal and F.Wilczek, Nucl.Phys. B399 (1993) 395.

3. C. De Tar, in these proceedings.